I was in Tecate last weekend.
1) Yes, there is parking on the American side. There is a dirt lot on the right, about 50 yards before the crossing. The driveway can be easy to miss. If you get to the tiny little strip shopping center at the crossing itself, you've gone too far. There is a trailer set back from the street that is visible from the road, the driveway to the parking lot is in front of that trailer, which also happens to be where you pay to park. You can either stop on the way in and pay, or go ahead and find a parking space and then pay as you walk out. The fee is $5 and the staff speaks both Spanish and English.
2) As for the food...if you're looking for something that is not a taqueria type place, you've probably got 3 options for "nice". Asao, Salmon, or El Lugar de Nos.
Asao is walkable (about 10 mins.) from the border. The facility is beautiful, the bar and patio are comfortable and cozy and the dining room has stunning views of the surrounding hills (which unfortunately also contain the eyesore that is the border fence). I've had very, very good meals here and I've also had some that were just "okay". It all depends upon who is cooking in the kitchen on any given day. If walking, you will pass right by El Pan Mejor de Tecate, one of the better panaderias anywhere in Mexico. Stop on the way back and bring some pastries home :-)
El Salmon is on Juarez. Here is the link to their Facebook page - https://www.facebook.com/ElSalmonTecate
The woman that runs it used to work at Rancho la Puerta and the food is very good
El Lugar de Nos is probably the most interesting option in Tecate at the moment. Everything they serve is grown or produced locally. They're currently highlighting the goat cheese from Valle 34. The cheese making venture began when the family got laid off from their jobs and the husband bought some goats. The wife and daughters began making goat cheese by trial and error, but they were successful. Successful enough to begin to sell their cheese locally. I an not a huge fan of goat cheese, but this is some pretty good stuff. The queso presando (pressed cheese) usually served with bread and marmelade is very good. Iv'e eaten at El Lugar de Nos multiple times. For the most part it is good to very good. https://www.facebook.com/LugarDeNos
You need to be aware that El Lugar de Nos also uses part of their venue to cater parties. If they've got a large catering function happening, they sometimes close the restaurant. They are open 1 pm - 1 am, I have only been there for dinner. I think their menu is the same all day, but I am not sure how the restaurant operates, or how good the food is, or is not, earlier in the day.
Unlike Tijuana, there are not a fleet of taxis waiting just as you walk across the border. Once you cross, if there is no taxi, and you need one, just wait on the curb across the street from the entry point and one will probably be along shortly.
Enjoy the day trip. Tecate is a little hidden gem.