The Hastyayurveda is a Sanskrit text that deals with the treatment of elephants. The author of the Hastyayurveda is attributed to Palakapya.

Option 1: Palakapya - This is the correct answer. Palakapya is believed to be the author of the Hastyayurveda, which focuses on the health and medical treatment of elephants.

Option 2: Vagbhata - Vagbhata is a renowned ancient Ayurvedic physician, but he is not associated with the authorship of the Hastyayurveda.

Option 3: Sushruta - Sushruta is considered to be the author of the Sushruta Samhita, one of the foundational texts of Ayurveda, but not the Hastyayurveda.

Option 4: Charaka - Charaka is the author of the Charaka Samhita, another important Ayurvedic text, but he is not associated with the Hastyayurveda.

In conclusion, Palakapya is the correct author of the Hastyayurveda.