There aren't a lot of restaurants in East LA I would take my parents, though I love East LA. Most of my favorites are holes in the wall that concentrate on food, not fun. La Parilla is a welcome exception.
Situated right in the middle of Avenida Cesar Chavez in Boyle Heights, La Parilla has historic roots in this heart of East LA. Strolling down this commercial strip, you'll realize the distance from LA to Mexico City is pretty small. Enjoy walking around during the day (it gets a little dicey at night) and seeing the fresh fruit vendors, people selling stuff on the street, cowboy boots in the windows, and the thumping ranchera music blasting from cars.
You can tell La Parilla is the jewel of the block. The waitresses bustle about in folklorico dresses, the booths are filled with cops and local businessmen on lunch breaks, and the walls are bursting with kitschy rural Mexican trinkets. They are not taken aback to the see the occasional gringo, and speak English and Spanish well.
The first sign that the food is great is that they have an old brick stove that is used for making fresh real handmade tortillas. If you have never had real tortillas, you don't know what you're missing. This is not a burrito/taco place (although I think they are on the menu). You order real Mexican classics. They have menudo on the weekends, chile rellenos, enchiladas, bistec, dorados, etc. I recommend their carne asada plate - a simple meal of Mexican rice, beans, a grilled green onion, and the perfect tender grilled flank steak. It's served with some fresh tortillas and homemade salsa. Unbelievable.
Enjoy the best of Boyle Heights!