This mission will be privately funded. Yes, SpaceX does receive quite a lot of money from NASA, but that is in payment for useful missions on Falcon 9 and with Dragon capsules. SpaceX is spending their own money, including profits from NASA missions and from Starlink, to find the Starship program.

With that being said, NASA is spending some money on Starship, but that is specifically for its development into the HLS moon lander, not for Mars missions.