1- "a place to park" most of the times means a parking, some building.
2- "a room to park" refers to a real room, something like a garage. (not the room as space)
3- "a space to park" is what you find in the corner of a street or in any other place.
But I think you can say:"There isn't much room in the yard to park a car." Probabely it's different from "find a room to park"
This example is mine but you can find some interesting examples in Meriam-Webster dictionary for room and space.
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