Thanks to Wikipedia: Parrilla (note different spelling) is the Spanish word for grill, and is sometimes used as a synonym for the Asado barbecue meats popular in Argentina and Uruguay, the type of restaurant to dine on these meats, and the area built in a home to cook them.
The parrilla is also a method of torture that takes its name from the grill or barbecue described above.
I use it in the sense of "mixed grill", which is common enough in Mexican/Central and South American restaurants, but usually disappointing and sometimes indeed tortuous. Ah ... for a good Argentinean or Brazilian mixed grill in Austin!
I've never tried it, but last time I was at Manuel's I saw they offer one, I think it's Sunday and Monday evenings. Just for the record, I've had ok luck with Manuel's apps, but not so much with some tasteless main courses, most recently an extremely bland shrimp dish.
re: Fatty Fryer
There's a parrilla up pn N Lamar in the Rundberg area. Haven't tried it yet. Sabor had a parrillada platter that was very good, but they are closed now.
Dona Emilia's has a veggie one that's been reported as good.
La Casita on Anderson, near Shoal Creek, has a parrillada section on their menu.