I don't know the fees but I'm sure they're high.

However I do know there is a public parking structure directly across from the Hotel Eden (on the south east side of Via Ludovisi and Via di Porta Pinciana). I had stayed at the Excelsior once and a guest had his car valet parked while visiting me - and I believe the Excelsior uses this lot (the Excelsior is at the corner of Via Ludovisi and Via Veneto). I would imagine the Eden uses this lot too. Maybe if you accessed this lot on your own - as opposed to using the valets, it might be cheaper?

I'll see if I can find some more info on this parking structure and I'll post again if I come with the cost...