Which branches of Buddhism acknowledge the Parable Sutra?

According to the Theravada, the authentic sermons of the Buddha are known as suttas, not sutras. However, in the Mahayana branch, there is a body of sutras known as the Agamas. These are Chinese translations of Sanskrit translations (now lost) of the Buddha's suttas. I do not know if the Parable Sutra is from the Agamas (which are historically-based) or Mahayana (not historically-based) sutras. I looked it over briefly now, and I did not notice anything that marked it as being from the Mahayana Sutras, so it may have a Theravadin equivalent.

Anything in the Taisho Tripitika is believed by the Mahayana to be the words of the Buddha.

When a Sutra began with "佛说" (佛说譬喻经), it usually means remembering what Buddha said and retold by a third party. This a general understanding, but doesn't undermine the value of teaching contained in that Sutra.

This 《佛说譬喻经》 only one page, translated by Yijing (635–713 CE), is collected in the Jātaka Section. Some scholars and Buddhists doubt the authenticity of the Jātaka "tales" - also in the Theravadin Canon. The Chinese Canon captioned this Sutra 佛说 as the beginning, just reflected the meticulousness of the ancient Chinese Buddhist Masters, to differentiate if that 100% confirmed directly from the official Tripitaka collecting, or the source was with certain uncertainty. Captioned with "佛/说" (Buddha/Said), means it told by someone that the Buddha said this. However, judging the Sutra context and other evidences, they aligned with the Buddhist teaching thus accepted in the Canon.

For Theravada, they have only one section out of the Twelves Sections of the Buddha Sutras. They may unlikely can have this in their canon. The Three Turning of Twelves Dharma Wheels was recounted in event of 500 Arhats collecting the Tripitaka after Buddha in Nirvana, recorded in at least all the survived Vinayas in the Chinese Canon, more detail in this post. The Mahisasakas Pratimoksa, one of the Early Eighteen Buddhist Schools, appeared in Ceylon supposedly split from Sthavira, but kept in the Chinese Canon. Surprised this Vinaya completely missing in the Pali Canon; their Vinaya Pitaka is with unknown origin.

Due to their lack of writing material to record large quantity of Sutras, only mean was relied on oral memorizing; Pali wasn't formally invented and synthesized, Theravada Suttas when finally they had the mean to write them down, was first written in Sinhalese letters imitating the "Pali" (Buddha never spoke Pali; Pali wasn't any ethnic language spoken by any race, the language existed and spoken by the ancient was called Paiśācī, and Prakrit, etc., not Pali; neither Pali appeared in the Ashoka Edicts, nor found to be used in any survived literature works apart from the Pali Canon) until Buddhaghosa translated all their Sinhalese records to Pali -- they likely incapable of holding this Sutra in their collection. Let alone if from the island they had means to access to all Sutras that circulating in the Indian Penisula, also wars, at that time. The only uncertain event recorded the Buddhist teaching arriving the island Ceylon was the story of Mahinda, even their own Dipavamsa and Mahavamsa had conflicting accounts on this story. If this story to a certain degree credible, how many Sutras could be bought by just one time of missionary effort?

Therefore it's not surprised this Sutra if not existed in the Theravada Canon, and not surprised many many Sutras also missing in the Theravada Canon. Just one Chinese Agama Section has the complete and extra Sutras that covered the totality of Sutta Pitaka of Pali Canon capable of keeping.