Are you staying in the hotel? If yes, hotel stuff will most certainly know of any nearby parkings.
I stayed in Casa Dona Elvira B&B on Padre Billini (great, clean, quiet place with WiFi and LAN) and there is an overnight parking just around the corner from this hotel (RD$60/night if you use it for a week and negotiate, RD$120/night if you pay what they ask for), walking distance from El Conde. Sorry I forgot the street addres for the parking, but just knock on the door of Dona Elvira hotel at 207 Padre Billini and ask for the overnight parking.
Found a map,
https://www.dona-elvira.com/Map.htm - the parking is on Jose Reyes street, very close to the corner on the left side, if you drive from Padre Billini, and two blocks from El Conde. There is no sign, just enter through the open gate into large parking lot.