[TexasGator]

The Jockey Club may issue a ‘racing permit’ for non-thoroughbreds that is accepted by state racing commissions.

I am sure they are very selective in granting this but I don’t know the details.

[ml/nj]

for non-thoroughbreds

This is the key phrase in what you posted. Some jurisdictions run Arabians, and maybe other breeds, I don't know, in separate races for that/those breed(s). But thoroughbreds only race with other thoroughbreds at any racetrack that ever runs a race in front of more than 200 people.

ML/NJ