Why didn't Caesar move against Sextus Pompey immediately after Munda?

Caesar did not depart "for Parthia". Although that was later embellished to be on his table as plans, it is not what he did.

He returned to Rome to hold an "unpopular triumph", after he defeated his main opponents in name (Labienus, Gnaeus) and he did seek out Pompeius's sons, 'pacifying the land', destroying the places where he believed them to hide. This must have taken him quite some time durign which he was kept quite busy with other things than 'let's start invading Parthia'.

When Caesar left for Rome the senate's fleet was essentially no more, Cato, Pompeius, Labienus, Gnaeus all dead and the forces in terms of man power not only significantly reduced but dispersed. From that view his opposition did no longer exist, at least for the time being, as we know now.

The immediate threat thus seemingly eliminated completely he consolidated his affairs in Rome – to become dictator in perpetuum and perhaps even striving for the title rex for the next campaign, finally again directed at an outward enemy.

That is of high significance. After four years of constant campaigning his troops were getting exhausted and wary and weary, needing a rest from the everlasting fighting, as evidenced by the legionaries behaviour already at Thapsus. His Gallic veterans therefore largely absent in Munda, he even had to 'fight for his life' at Munda and won with a big chunk of luck.
Civil war means Romans fighting Romans, not very attractive when you know that fighting non-Romans means plenty of booty.

It is therefore asking with a lot of hindsight and an expectational reading of Caesar's character to be merciless and eliminatory to wonder why he didn't hunt down Sextus. When Caesar left

Sextus Pompeius remained at large, after Munda there were no more conservative armies challenging Caesar's dominion. Upon his return to Rome, according to Plutarch, the "triumph which he celebrated for this victory displeased the Romans beyond any thing. For he had not defeated foreign generals, or barbarian kings, but had destroyed the children and family of one of the greatest men of Rome."

It may remain debatable whether like in Thapsus Caesar would have liked to show clementia Caesaris towards Sextus, like he wanted to treat Cato, or whether he would eventually direct more focus on that youngish fugitive person called Sextus. The sources can even sustain to suggest that Sextus did cause so much trouble immediately that he was the reason for Caesar's unusually long return trip to Rome.

Alternatively, Sextus Pompeius is seen as an isolated entity, without allies or explanation. […]
Giving Sextus Pompeius his due necessitates a reassessment of two key elements of the accepted narrative. In the first place, we must face the fact of continuous military activity from 49 to 30 rather than two separate phases separated by a period of peace between 45 and 43. During that time, Pompeius inflicted more than one defeat upon the elder Caesar’s generals and in doing so clawed back control of the Iberian peninsula. Munda did not mark ‘the end’ any more than Pharsalus in 48 or Thapsus in 46. Had Caesar survived, he would have had to pay closer attention to Sextus Pompeius than the unpopular triumph at the end of 45 was designed to indicate. […]

The elder Caesar displayed the goddess Pietas on his coinage in 48 (RRC 450; Gelzer 1968, 201) and accepted the honour of the corona civica after his victory over Gnaeus Pompeius at Munda (App. BC 2.104—106). It is clementia and humanitas that dominate the rhetoric of the early forties, in reaction to the savagery of Sulla and Marius four decades earlier; however claims of pietas form an important subtext to the clementia discourse.

Having lost it around 5 March, Gnaeus fell back to the town of Urso, a short distance from Munda. On 17 March, the two armies met at Munda in a bloody climax to a bloody campaign.

Only Appian (BC 2.106) says that Caesar hurried back to Rome after the victory. He must have taken many months to return. Nicolaus (/r. 127.10) places his arrival in September and Velleius October (2.56.3). Dio (43.39.4—5) does not specify a month but, along with Suetonius (Jul. 42.1), says that Caesar was kept busy with a settlement programme, which also implies a return to Rome in the autumn. Believing that the war was over, Caesar also withdrew the majority of his army, leaving the fate of Sextus Pompeius in the hands of local allies, until matters once again became so pressing that a new force had to be sent to deal with it (Dio 45.10; Lowe 2002, 67). […]

For example, Appian does not refer to Pompeius between the moment he breaks off from the Iberian campaign {BC 2.105-6) and his misplaced notice concerning Antonius’interest in him at the beginning of 44 {BC 3.4). Worse still, Florus omits reference to him between the events of 45 (2.13.8) and the Bellum Siculum of 38 (2.18.7). Dio mentions him at 43.39.1, then re-introduces him at 45.10 (in the context of events at the end of 44) and then again at 48.17. For Velleius, Sextus Pompeius hardly exists until after the battle of Philippi.25Plutarch omits him until relating the events of 39 {Ant. 32).26Gowing (1992,183) argues that Dio is more culpable in his arrangement of the material concerning Sextus Pompeius than Appian, but the tendency to provide minimal or disarticulated detail, on the activities and importance of Sextus Pompeius between 45 and 38, is universal.

Appian’s choice of the verb 'demolish' to describe the end of ‘all the civil wars’ reflects the fashion for seeing Munda as a point of closure, just as ‘practise piracy’ to describe Pompeius’ lifestyle relegates him to the status of brigand. However, at the beginning of BC2.106, the first clause (‘But he on the one hand still…') balances (‘Caesar, on the other hand … hurried’). Carter (1996, 125) chose to place the full stop after (‘lived by raiding’) instead of (‘being called’), thus neatly tucking Pompeius into the Iberian narrative and allowing Caesar to return to Rome unencumbered. And that is not what Appian said. […]

Indeed, Sextus Pompeius continued to matter. After he escaped from Corduba, he made his way to the north western region of Lacetania and from his base in the north-western area of the peninsula he gathered a vast number of legionaries and allies.28It was not until October that Caesar himself could leave the Iberian peninsula, and by then it was already obvious that Pompeius, like the other Republican commanders since Pharsalus, was going to keep fighting. His army consisted of the Roman soldiers who survived the battle (and several other campaigns) along with the many Hispanic tribes who remained loyal to his father’s memory. Arabio, an African king who had suffered at the hands of the Caesarian Sittius, also made his way to Pompeius’ camp from his territory in Africa.29 Even as Caesar celebrated a triumph over the brothers, one of them was already reinvigorating the conflict (Schor 1978, 33). Before the end of the year, Gaius Carrinas had been sent to Hispania Ulterior after Caesar returned to Italy, but he was unable to make any headway (App. B C 4.83). Caesar then dispatched a second commander, Pollio, who was as unsuccessful as Carrinas. […]

Cassius’ attitude to Gnaeus Pompeius was especially legendary. The elder Seneca ('Suas, 1.5) quotes the letter from memory, substituting stultitia for crudelitas. Such a rejection served to divest the survivors of Thapsus of the respectable title of defender of the respublica, whatever they themselves might have believed they were doing. In the dark period after Munda, Sextus Pompeius had litde chance of winning any ideological debates. The losses his side had incurred at Munda and afterwards left him without even the legatipro praetore V arus and Labienus to bestow a vestige o f an institutionally sanctioned chain o f command. In a five-year period, Caesar had eliminated or disengaged all holders of Imperium from the Republican camp. If even one had remained, it would have been harder (though not impossible) to characterise the survivors as a band of pirates. […]

When Sextus Pompeius had left Rome with his father in 49, Gnaeus Pompeius Magnus had been the wealthiest and most powerful Roman the city had ever known. At the end of 45, Sextus Pompeius was fighting for his existence, relying completely on his own wits and the good will his father’s name still generated. Although preserved from the Pharsalus campaign, he had been a witness to all subsequent events of the war as he had been to the brutal death of his father at the hands of the soldiers of the Alexandrian court. Yet because of the carnage of Munda, following as it did the deaths of Scipio, Cato and the leaders of the Republican camp in Africa, he could be made to look like an isolated brigand, completely cut off from any association with Rome and the cause which had led his father to fight in the first place.

_{–– Kathryn Welch: "Sextus Pompeius and the Transformation of the Roman Republic", The Classical Press of Wales: Swansea, 2012.}

In all, after Munda Sextus enjoyed some precariously small sucesses in a dire situation:

With the company of among others Titus Labienus he was able to reach Spain (Dio Cass. 43.30.4), where he immediately entered on the duties of commander of the garrison of Corduba (B. Hisp. 3.1). It is there where the news of his brother’s defeat at Munda in the March 17, 45 BC reached him. He left the city soon after and went north, where he hid himself among the Lacetanii (Strabo 3.4.10; Dio Cass. 45.10.1) and then the Celtiberians (Flor. 2.13.87). During the time he spent there he started recruiting new forces and engaged in guerrilla warfare at once (App. B Civ. 4.83; Dio Cass. 45.10.2). Shortly afterwards he moved to the south and captured several cities, like his brother had done. But it was the death of Caesar that gave him a true free hand (Dio Cass. 45.10.3). Probably around the time of his return to the south he started to mint his own coins (Hadas 1966, 42-44, 49-52; Amela Valverde 2001, 13-17, 23-25; Lowe 2002, 13-17; Amela Valverde 2002, 44-45, 52-53).
_{–– Kamil Kopij: "Pietas in the Propaganda of Sextus Pompey", in: Ewdoksia Papuci-Władyka (Ed): "Studies in Ancient Art and Civilization 15", Kraków, 2011.}

Even after Gnaeus was defeated and killed by Caesar’s forces at the Battle of Munda in 45 B.C.E., Sextus Pompey remained in Spain and began a course of “guerilla warfare,” winning a number of small yet decisive victories over Caesarian governors in the area (Gabba 155).4 He took refuge in the area known as Lacetania, surviving and eluding discovery mainly because of the kind disposition of the natives toward him, due to their reverence for the memory of his father (Cass. Dio 45.10.1).

Even so, it is possible that Sextus’ eccentric military actions in Spain at this time fueled later criticism of him as a “pirate” and rebel against the Second Triumvirate. Regardless of these later opinions, however, many of Sextus’ contemporaries saw his recurrent struggle against Caesarian rule in Spain as familial vengeance, and not necessarily as lawlessness. In a way, even though all hope of defeating Caesar was lost, Sextus probably envisioned himself as the participant in a righteous family feud with the Caesarians.
_{–– Kate Rogers: "Sextus Pompeius: Rebellious Pirate or Imitative Son?", Chrestomathy: Annual Review of Undergraduate Research, School of Humanities and Social Sciences, School of Languages, Cultures, and World Affairs, College of Charleston Volume 7, 2008: pp. 199-226.}

Question:
Why didn't Caesar move against Sextus Pompey immediately after Munda?

Caesar did pursue Sextus. After the Battle of Munda(17 March 45 B.C) both of Pompey's son's Gnaeus Pompeius (oldest son) and Sextus Pompey(youngest son) ran for their lives. Pompeian armies had been destroyed, their supporters had been exhausted, and Caesar was the clear winner of the Civil war. Gnaeus Pompeius was caught and executed a few weeks later, 12 April 45 BC. Sextus was hunted but was able to stay one step ahead of his pursuers.

Gnaeus Pompeius
(After The Battle of Munda) - Gnaeus and Sextus managed to escape another time but supporters were difficult to find. It was by now clear Caesar had won the civil war. Within a few weeks, Gnaeus Pompeius was cornered and killed by Lucius Caesennius Lento. Sextus Pompeius was able to keep one step ahead of his enemies, and survived his brother for another decade.

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The Battle of Munda
Although Sextus Pompeius remained at large, after Munda there were no more conservative armies challenging Caesar's dominion.

Sextus did not rebuild his forces and renew his armed opposition to the new order until after Julius Caesar had been killed, after the second Triumvirate was formed, and after the second triumvirate had dealt with Brutus and Cassius. In other words Sextus was an afterthought at this point. He only formed his army because Agustus, Antonius and Lepodus made it clear they were coming for Sextus after they dealt with those who had slain Caesar.

Sextus Pompey The Second Triumvirate was formed by Gaius Julius Caesar Octavianus, Marcus Antonius and Marcus Aemelius Lepidus, with the intention of avenging Caesar and subduing all opposition. Sextus Pompeius in Sicily was certainly a rebellious man, but the faction of Cassius and Brutus was the second triumvirate's first priority. Thus, with the whole island as his base, Sextus had the time and resources to develop an army and, even more importantly, a strong navy operated by Sicilian marines.

I subscribe to the simpler explanation: Sextus Pompey was overshadowed by his elder brother Gnaues Pompeius (Pompey the Younger) and, of course, their father - Pompey (Pompey the Great). Until their passing, the decisions and actions taken by this family were always attributed to Sextus' elder brother and father.

So, I think your "possible answers" really is the most plausible, and generally accepted, answer in that Sextus Pompeius was, first, not recognised as a significant military leader until he stepped up or into his brother's shoes after the Battle of Munda (March 45 BCE).

Second, he wasn't the bigger voice in opposition to the triumvirs until 42 BCE. That would be Brutus (Brutus the Younger) and Cassius. Both of them died during the Battle of Philippi (3/23 Oct 42 BCE).

Only then would Sextus be seen as having a larger role, i.e. "given space to shine". By this time (42 BCE), Caesar was already dead (2 years earlier).

Kathry Welch's "Magnus Pius: Sextus Pompeius and the Transformation of the Roman Republic" (2012) is a good book on this point, that Sextus' role was always under-appreciated by Roman leaders (e.g. Caesar and Octavius) and later historians.