1) In Bala Kanda, Srimad Ramayana he was described as the son of Aristhanemi.
अरिष्ठनेमि दुहिता सुपर्ण भगिनी तु सा | द्वितीया सगरस्य आसीत् पत्नी सुमति संज्ञिता || (Bala Kanda 38 Sarga 4 Sloka)
"The daughter of Arishtanemi, and who is also the sister of Garuda, the Divine-Eagle, is the second wife of Sagara, who is known by the name Sumati".
Here, Suparna (सुपर्ण ) indicates the one having beautiful feathers, i.e., Garuda.
2) Further, while describing Sri Hanuma's power and celerity, Jambavanta equals his speed with that of Garuda, as follows:
अरिष्टनेमिनः पुत्रो वैनतेयो महाबलः | गरुत्मान् इव विख्यात उत्तमः सर्व पक्षिणाम् || (Kishkinda kanda 66 Sarga 4th Sloka)
"The son of Aristanemi and Lady Vinata is the best bird among all the birds, who is superbly mighty and who is universally renowned as Garuda."
पक्षयोः यत् बलम् तस्य तावत् भुज बलम् तव | विक्रमः च अपि वेगः च न ते तेन अपहीयते || (Kishkinda kanda 66 Sarga 6th Sloka)
"Whatever is the strength of his wings, that much is the strength of you arms, and even by the yardsticks of his dash and dare, yours too, are unreprovable".
3) On his first meeting with Sri Rama, Jatayu, a great eagle and a friend of Dasaratha, narrates his lineage.
कश्यपः प्रतिजग्राह तासाम् अष्टौ सुमध्यमाः | अदितिम् च दितिम् चैव दनूम् अपि च कालकाम् || ताम्राम् क्रोध वशाम् चैव मनुम् च अप्य् अनलाम् अपि | (Aranya Kanda 14 Sarga 11-12 Slokas)
"Of them Kashyapa accepted eight slender-waisted daughters of Daksha Prajapati, namely Aditi, Diti, Danu, Kaalakaa and Taamra, Krodhavasha, also thus Manu and even Anala also as wives.
क्रौन्चीम् भासीम् तथा श्येनीम् धृतराष्ट्रीम् तथा शुकीम् || ताम्रा तु सुषुवे कन्याः पंच एता लोकविश्रुताः | (Aranya Kanda 14 Sarga 17-18 Slokas)
"But Taamraa gave birth to five world renowned girls namely, Krounchi, Bhaasii, Shyenii, Dhritaraashtrii, and Shukii.
चक्रवाकाम् च भद्रम् ते विजज्ञे सा अपि भामिनी | शुकी नताम् विजज्ञे तु नताया विनता सुता || (Aranya Kanda 14 Sarga 20 Sloka)
"Dhritaraashtrii gave birth even to Chakravaaka water-birds, and Shukii gave birth to a daughter Nata, and Nata's daughter is Vinata."
Hence, Vinata was the Great Grand-daughter of Prajapati Kashyapa ( Kashyapa + Taamra -> Shukii -> Nata -> Vinata) and Garuda was the son of Vinata.
According to this narration, Garuda was the son of Aristhanemi. However, Mahabharata says that he was the son of Kashyapa.
Which one is correct?