There is no nearby parking space left around this house.
Does parking room or parking spot sound correctly if it is used as a synonym in above sentence?
What is the synonym to use here if they are both not accurate?
No I wouldn't use parking room.
parking spaces (plural) is probably OK, but more likely refers to a place with marked parking spots, usually in a business district or a parking lot.
There were no parking spaces near the store.
When referring to parking in general, you can simply say
There is no nearby parking left around this house.
parking is understood to be a place to park a motor vehicle.
parking spot refers to a specific place to park one vehicle, but I can't say if it applies to your situation.
There is a parking spot next to the fire hydrant.