The father of Shandilya is mentioned as Devala, according to the Saurapurana:
Kaśyapa created the animals, animates and inanimates and again for the growth of progeny he underwent austere penance. By the power of penance two sons namely Vatsara and Asita were born. Devala muni was the son of Asita born of Ekaparṇā. Devala attained great perfection by worshipping Śambhu. From Devala was born Śāṇḍilya. These are the progeny of Kaśyapa.
The Shandilya-gotra is named so because "fire" was born (incarnated?) into this lineage, according to the Puranic Encyclopedia
Śāṇḍilya—A maharṣi born in the dynasty of Kaśyapa, son of Marīci. As Agni was born in the family of the maharṣi it came to be called 'Śāṇḍilyagotrīya' (born in the family of Śāṇḍilya). King Sumanyu once gave him food and other edible things. (Anuśāsana Parva, Chapter 137, Verse 22).
Another abstract passage mentions Shandilya in connection with fire, in the Brahmanda Purana:
91-92. Formerly, in his Yajña this Gāthā (verse) was sung by the great sages after seeing the birth, from fire, of the noble-souled Śāṇḍilya—“Those men who see the Yajamāna (the sponsor of the sacrifice) Dilīpa with mental purity, are the conquerors of heaven. (That king Dilīpa) is noble-souled and is truthful in his observance of rites.”
Another interesting note is that the Skanda Purana mentions Likhita and Shankha as sons of Sage Shandilya, although the Brahmanda Purana (see link above) mentions this slightly different. It is also worth cross-referencing these two genealogical legends with the one from the Saura-purana, which supports the Skandapurana in several ancestral details.
A possible explanation for this is mentioned in the Vedic Index of names and Subjects, where Shandilya is said to be a patronym applied to several teachers.
Śāṇḍilya, —‘descendant of Śaṇḍila,’ is the patronymic of several teachers (see Udara and Suyajña). The most important Śāṇḍilya is the one cited several times as an authority in the Śatapatha Brāhmaṇa, where his Agni, or ‘sacrificial fire,’ is called Śāṇḍila. From this it appears clearly that he was one of the great teachers of the fire ritual which occupies the fifth and following books of the Śatapatha Brāhmaṇa. In the Vaṃśa (list of teachers) at the end of the tenth book he is given as a pupil of Kuśri and a teacher of Vātsya; another list at the end of the last book in the Kāṇva recension4 gives him as a pupil of Vātsya, and the latter as a pupil of Kuśri.