First of all, you're 0 for 20 because the Parthenon is in Athens, Greece.
But, if you're in Rome, almost all restaurants are open on Sundays - it's mondays that's the problem.
Try the Tuscan place in Piazza Pasquino (just off pzza navona)
My favorite in Trastevere (just across the bridge from Largo Argentina) on via della luce Da Carlone - great focaccia and the cacio pepe
And, a bunch of places around Piazza dell'orologio behind Chiesa Nuova.
A classic is Costanza's
There are actually quite a few places, including many of the places people complain are closed on Monday (though many are closed both days or at least Sun evening and Monday). Depends what you want but a few off the top of my head are Vecchia Roma (Piazza Campitelli), Campana, Grano, Pierluigi, Ditirambo, Il Sanlorenzo, Fiammetta, and all, or nearly, pizzerias and restaurants in hotels, some of which are very good.
in reviewing the available places, OP also needs to be clear what is meant by "Dinner" - for an american this can be either the midday or evening meal.
Sometimes places in Italy will be open at lunchtime and closed in the evening or vice versa, especially on sunday where lunch tends to be the big thing. the fact that so many places in Rome centro are closed on Sunday is proof that the restaurant business there is not wholly tourist driven!