Ayurveda is the science of life which is originated from the mukha (mouth) of Lord Brahma or some text also said that it is memorised by Brahma which further propagated to the mankind. So as per that technically the father of Ayurveda is Lord Brahma (the creator of universe). 

First of all Lord Brahma teaches this knowledge to Daksha Prajapati (lord of creatures)……….which further passes to Ashvini kumaras (twin God of medicine/ doctors of Gods)………..then it passes to Indra (king of Gods)…….after that he teaches it to Rishi Bhardwaja (enlightened person, who went to Indraloka in search of Ayurveda)………lastly he teaches this to Atreya Punarvasu.

Atreya Punarvasu (one of the elightened person who first get the knowledge from Rishi Bhardwaja) come to earth and propagate the Ayurveda with the help of its 6 students named as Agnivesha, Bhela, Jatukarna, Ksharapani, Harita & Parashara. They each wrote their own Samhita (compendium) on the Ayurveda. The most popular one is Charaka samhita which is based upon the Agnivesha samhita. It is available in present day also and also taught in all schools of Ayuveda even today.

All the above information is also taken from Charaka samhita itself. Technically Brahma is the father of Ayurveda.