Restaurants are more likely to be open, but some of them could be closed on Good Friday, too. Anyway, considering the enormous offer of them in Madrid, you should not have any problem to find one to fit your needs.
On those days there are processions rather than parades. Despite not being so traditional and spectacular in Madrid as in other parts of Spain (Andalusia, Valladolid, Zamora, Murcia,...) it can be interesting to watch. Most of them run on the old Madrid streets, so it will be very easy to find if you are in a centrally located hotel. Ask in the reception or catch any of the most popular newspapers (ABC, El Mundo, EL PAÍS,...) where you will find plans of their routes.