What happens to consciousness/awareness when entering Paranirvana?

The term "parinibbana" does not always refer to the ending of the life of an arahant. Parinibbana can also refer to here-&-now Nibbana, such as in SN 22.53, which uses the term or verb "parinibbāyati". SN 22.53 refers to nibbana in the here & now, which is the destruction of craving.

Nibbana is not a state of consciousness but is an object (ayatana) of consciousness (Ud 8.1).

"Birth" & "death" are self-views. Arahants are free from birth & death. For example, MN 140 says:

Bhikkhu, ‘I am’ is a conceiving; ‘I am this’ is a conceiving; ‘I shall be’ is a conceiving; ‘I shall not be’ is a conceiving; ‘I shall be possessed of form’ is a conceiving; ‘I shall be formless’ is a conceiving; ‘I shall be percipient’ is a conceiving; ‘I shall be non-percipient’ is a conceiving; ‘I shall be neither-percipient-nor-non-percipient’ is a conceiving. Conceiving is a disease, conceiving is a tumour, conceiving is a dart. By overcoming all conceivings, bhikkhu, one is called a sage at peace. And the sage at peace is not born, does not age, does not die; he is not shaken and does not yearn. For there is nothing present in him by which he might be born. Not being born, how could he age? Not ageing, how could he die? Not dying, how could he be shaken? Not being shaken, why should he yearn?

Enlightenment is pursued to end suffering; just as food is eaten to end the pain of hunger or a doctor is consulted when sick. Just because life eventually ends does not mean we stop eating food everyday. Similarly, just because life eventually ends does not mean we do seek real happiness & real peace in the here-&-now.

In summary, when life ends, consciousness, which is dependent upon the mind & body (SN 22.82), also ends (SN 22.85). Therefore, in the final Paranibbana of an Arahant or Buddha, as described in DN 16, the consciousness of the Arahant or Buddha ends, as described in SN 22.85.

But when Nibbana is attained in the here-&-now, consciousness does not end, as described in SN 22.53. The term "cessation" ("nirodha") does not mean the cessation of consciousness but means the cessation of ignorance & craving polluting consciousness. Refer to SN 22.5, which explains the meaning of "arising" ("samudaya") and thus "cessation"; as follows:

What is the origin of consciousness? Here, bhikkhus, one seeks delight, one welcomes, one remains holding.

And what, bhikkhus, is the passing away of consciousness?.... Here, bhikkhus, one does not seek delight, one does not welcome, one does not remain holding.

According to Iti 44:

And what is the Unbinding property with fuel remaining? There is the case where a monk is an arahant whose fermentations have ended, who has reached fulfillment, finished the task, laid down the burden, attained the true goal, destroyed the fetter of becoming, and is released through right gnosis. His five sense faculties still remain and, owing to their being intact, he is cognizant of the pleasant & the unpleasant, and is sensitive to pleasure & pain. His ending of passion, aversion, & delusion is termed the Unbinding property with fuel remaining.

And what is the Unbinding property with no fuel remaining? There is the case where a monk is an arahant whose fermentations have ended, who has reached fulfillment, finished the task, laid down the burden, attained the true goal, destroyed the fetter of becoming, and is released through right gnosis. For him, all that is sensed, being unrelished, will grow cold right here. This is termed the Unbinding property with no fuel remaining.

Ven. Thanissaro in his commentary of Iti 44, stated:

With fuel remaining (sa-upadisesa) and with no fuel remaining (anupadisesa): The analogy here is to a fire. In the first case, the flames are out, but the embers are still glowing. In the second, the fire is so thoroughly out that the embers have grown cold. The "fuel" here is the five aggregates. While the arahant is still alive, he/she still experiences the five aggregates, but they do not burn with the fires of passion, aversion, or delusion. When the arahant passes away, there is no longer any experience of aggregates here or anywhere else.

This means that consciousness ceases on parinirvana (i.e. when no fuel is remaining).

Nibbana with fuel remaining is when the arahant is living, and his five aggregates are still functioning, but they are no longer clinging aggregates (SN 22.48) i.e. they are no longer defiled.

So, living arahants still have sankhara. What they do not have is latent tendencies (anusaya), defilements (kilesa), effluents (asava), fetters (samyojana), craving (tanha) and clinging (upadana).

"Consciousness, thus not having landed, not increasing, not concocting, is released" (SN 22.53) simply refers to consciousness that is no longer defiled i.e. consciousness that is not a clinging aggregate - also please see this question. It does not mean that it is permanent. Upon parinirvana, it will cease - I'm referring to the Nibbana element without fuel remaining. Please also see this answer for details.

In MN 49, there was a "consciousness without surface" or "consciousness that is invisible, infinite, radiant all round" (viññāṇaṃ anidassanaṃ) that some people imagined to be a permanent consciousness of sorts, which turned out to be a mistranslation, according to the answers of this question. The sutta was talking about Nibbana. I've also explained this in this answer.

What about the notion that "their consciousness does not take rebirth into a physical form"? The idea of the consciousness of an enlightened one continuing to exist after death in some form without rebirth is not part of the teachings of the Early Buddhist Texts (EBTs).

I mean if everything what I am, even consciousness, which I think I ultimately am, vanishes, why should I pursue enlightenment then?

To end suffering.

Suffering pervades our lives, touching all that we relish. We spend our lives chasing the rebirth of past pleasures large and small. The first kiss leads to the second and then one day there are no more. This is the first noble truth, the truth of suffering.

The second truth is the origin of that suffering.

MN9:14-18.8: It’s the craving that leads to future rebirth, mixed up with relishing and greed, looking for enjoyment in various different realms. That is, craving for sensual pleasures, craving for continued existence, and craving to end existence.

Consciousness comes and goes. It is a burden to be borne, yet we cling to it as precious. In seeking continuity of consciousness there is suffering. Since consciousness arises, it must also die. Grasping at consciousness is therefore delusional, since it generates and perpetuates suffering.

"Everything what I am" is a lead weight. Why carry it?

Any form of existence after parinirvana is "an inconceivable" in the Buddha's Dharma. This is equally true of Theravada as it is of Mahayana here. The Buddha, "after death," is inconceivable. The Buddha's mind, after death, is inconceivable. The Buddha's mind, before death, is also inconceivable. The Buddha questions Vesālī:

“What do you think, Anurādha? Do you regard the Realized One as form?” “No, sir.” “Do you regard the Realized One as feeling … perception … choices … consciousness?” “No, sir.” “What do you think, Anurādha? Do you regard the Realized One as in form?” “No, sir.” “Or do you regard the Realized One as distinct from form?” “No, sir.” “Do you regard the Realized One as in feeling … or distinct from feeling … as in perception … or distinct from perception … as in choices … or distinct from choices … as in consciousness … or as distinct from consciousness?” “No, sir.” “What do you think, Anurādha? Do you regard the Realized One as possessing form, feeling, perception, choices, and consciousness?” “No, sir.” “What do you think, Anurādha? Do you regard the Realized One as one who is without form, feeling, perception, choices, and consciousness?” “No, sir.”

(Anurādhasutta SN 44.2)

The Buddha is not defined via five aggregates like a sattva is. This is before even parinirvāṇa. After parinirvāṇa, the Buddha is also not defined via five aggregates. With this in mind, we can go to the 25th Parīkṣā of the Mūlamadhyamakakārikā with the Madhyamakaśāstra commentary from ~2-400 AD by Venerable Vimalākṣa (青目, an alternate Sanskritization is "Piṅgala" )

25:17 The Tathagata, after parinirvāṇa, is neither said to exist nor not to exist, nor is he said to both exist and not exist, nor neither exist nor not exist.

25:18 The Tathagata, presently, is neither said to exist nor not to exist, nor is he said to both exist and not exist, nor neither exist nor not exist.

"Presently" here should be taken as "while alive."

25:19 Between nirvāṇa and the world, there is not the slightest differentiation. Between the world and nirvāṇa, there is also not the slightest differentiation.

25:20 From the true apex of nirvāṇa to the apex of the world, like this, there are two apices with not a sliver of difference between them.

25:21 "Existence or nonexistence after parinirvāṇa," et cetera, "the boundaries of existence," "constancy," et cetera -- all (of these) views depend on nirvāṇa being in the future or in the past.

Venerable Vimalākṣa comments:

(1) "The Tathagata after parinirvāṇa exists," or (2) "The Tathagata does not exist," or (3) "The Tathagata both exists and does not exist," or (4) "The Tathagata neither exists nor not exists," or (5) "The world is finite," or (6) "The world is infinite," or (7) "The world is both finite and infinite," or (8) "The world is neither finite nor infinite," or (9) "The world is constant," or (10) "The world is inconstant," or (11) "The world is both constant and inconstant," or (12) "The world is neither constant nor inconstant" -- these in three kinds are twelve views. When after the Tathagata's parinirvāṇa he is (1) existent or (2) nonexistent, et cetera (3-4), those four views arise dependent on (the notion of) nirvāṇa. When the world is (5) finite or (6) infinite, et cetera (7-8), those four views arise dependent on (the notion of) the future. When the world is (9) constant or (10) inconstant, et cetera (11-12), those four views are dependent on (the notion of) the past. Whether the Tathagata after parinirvāṇa exists or does not exist, et cetera, is inconceivable, and nirvāṇa is also the same. Like the origin and destiny of the world, if it is finite, infinite, constant, inconstant, et cetera, it is inconceivable, and nirvāṇa is also the same. Consequently, we say that the world and nirvāṇa are not different.

Āyuṣmat Nāgārjuna's Kārikā continues.

25:22 If all phenomena are empty, what is finite, infinite, both finite and infinite, or neither finite nor infinite?

25:23 Why is there the one and the many? Why is there the constant, and inconstant, and both constant and inconstant, and neither constant nor inconstant?

25:24 All phenomena being inconceivable is the cessation of the frivolous ponderings. To no person and in no place has the Buddha ever spoken.

Venerable Vimalākṣa comments again:

All phenomena at all times of every variety conform to dependent origination. All in all, they are empty, and thus they have not their own natures (自性, svabhāva). Which body exists to be the same as the soul? Which body exists to be different than the soul? Thus proceeding are the sixty-two demonic views. Each and every one within emptiness is untenable. When all existence completely ceases, frivolous ponderings are entirely gone. When frivolous ponderings are entirely gone, it is because we have penetrated the true aspect of the many phenomena and attained the tranquil path. Hearkening back to the chapter on causality, if we inquire into the many phenomena, they are neither existent nor nonexistent, nor are they both existent and nonexistent, and nor are they neither existent nor nonexistent. This is called 'the true aspect of all phenomena.' It is also called 'the true nature of the phenomena,' 'reality,' and 'nirvāṇa.' Therefore, the Tathagata at no time, in no place, to no persons, ever spoke of nirvāṇa as with particularized characteristics. When all existence has entirely come to an end, frivolous ponderings have ceased.

(Madhyamakaśāstra T1564.34c14)

Which directs us back to the OP inquiry: What happens to consciousness when entering parinirvāṇa? It ceases.

Of all forms of disease, none is greater than that of having a body — he extinguished the body. Of all forms of torment, none is more severe than that of having a calculating mind — he erased it and submerged in the vacuous. The mind is taxed by the body; the body is burdened by the intellect. The two pull each other, turning like a wheel on the endless road of misery. It is said in the sutra, “The intellect is poison, the body is shackles. Because of them the quiet silence of liberation remains beyond reach; they are the cause of all tribulations.”

The Perfect turned his body into ashes and extinguished his intellect, he relinquished his form and discarded his reason. Within, he abandoned the stirrings of illumination; without, he put to rest the basis of misery. Transcendent, perfectly free from all existents; boundless, he became great and vacuous. Tranquil, inaudible, clear, non-manifest, mysteriously gone forever into a destination unknown. When a lamp goes out its flame is extinguished, the oil and the flame gone all at once.

This is nirvana without remainder. In the words of a sutra, “The five aggregates are no more, like a flame extinguished.”

[...]

The terms “with remainder” and “without remainder” are only external appellations for nirvana, conventional designations for the divergent modes of the sages’ responding to things.

[...]

As for these (two) appellations (of "with remainder" and "without remainder"), they are established to indicate the various modes of sagely response. When the Sage manifests traces, we call this “arising”; when he makes them vanish, we call this “cessation.” His “arising” is referred to as nirvana “with remainder,” his “cessation” as nirvana “without remainder.” All along both appellations — “with” and “without remainder” — remain rooted in the nameless. Surely the nameless will take any name? Thus, the Perfect becomes a square when he inhabits a square, a circle when he stops in a circle, a deva when among devas, a human being when among humans. To become a deva or a human being in accordance with circumstance is surely not something that devas or humans could do. It is precisely because he is neither a deva nor a human that he can become one or the other.

(Venerable Sēngzhào Zhào's Essays T1858, translator unknown by poster, published by BDK as Three Short Treatises by Vasubandhu, Sengzhao, and Zongmi)

Though the Buddha's mind ended upon parinirvāṇa, he was never identified as, with, or in his mind to begin with. When I say "to begin with," I mean "after the bodhimaṇḍa," after he touched the earth in response to Mara, the godling prince. Nothing changes for the Tathāgata when form, consciousness, etc., ends, because he has already discarded, transcended, and escaped all of these.

In the Sarvāstivādin Mahāvibhāṣa ("The Great Commentary" of the Sarvāstivādin Śrāvakas), it explains nirvāṇa using some clever wordplay: "Vāna" means forest and "nir" means escape. As it is the escape from the forest of the aggregates, it is called nirvana.

When we ask a question like "Is the Buddha aware after parinirvāṇa," we are trying to pin the Tathāgata down, to identify him, using the aggregates by which ignorant sentient beings are bound. This can't be done.

Overlong, but hopefully helpful in some way.