What ever you do, do NOT park in one of the shopping mall lots--Las Americans, K-Mart, etc.--as your car will be towed. There is (sometimes) free parking that can be found on the public streets, but I recommend against using it.
I once had a car stolen that was parked on a US street near the border, but I've never had a problem using a paid lot with a 24-hour attendant. I'd advise against parking further north and taking the trolley to the border as your car is much safer using a lot near the border with a 24-hour attendant than it is being left in an unattended park and ride lot.
Use one of the pay lots; one with a 24-hour attendant. Some lots charge in 8-hour increments, some (such as Border Station) "re-up" at 10am, some (such as UTEP) give a true 24-hour rate (for around $9.00). Since you're planning on staying over night I think the UTEP parking lot next to the Duty Free store would be the best lot to use.
I don't recommend "newbies" drive in Tijuana. You're better off learning your way around TJ before you start driving in Tijuana. The return trip by car is often an hour (make that 2 hours on a Saturday) compared to the much shorter time pedestrians spend crossing the border. There is a shortage of street parking in TJ, plus it's much safer to use a paid TJ lot (which often charge more than US lots). Driving in TJ is very different than driving in the US (in the beginning it frightens most people) and driving at night when the cops and drunks are active is never fun.
I recommend doing what most people who are going to attend the fiesta do: take a taxi. It's $5.00 each way. (Mexico is a third world country. If you're the type that's going to "freak out" because Tijuana taxis are "too dirty" then you really shouldn't be in Mexico. Also, for what it's worth, in my experiences taxis in the US are just as dirty--and clean--as those in TJ.)
Be sure you bring the necessary documentation--green card, passport, etc.--to return to the U.S. With you "paperwork" in order you won't have any problems.
Be sure to say "Hi" at the fiesta! (I'm a fairly well known "commodity," so if you ask around you'll find me.)