Yes. It should be --- Park Road.
No because it is not a proper nou n.
Amusement park is only capital if it is the name of a location. For instance, you would capitalize Cedar Point Amusement Park. Amusement park by itself, however, is just a noun, and only follows standard sentence grammar (Capitalize amusement if it's the first word in a sentence, etc.)
The New Hyde Park Road School is on New Hyde Park Road, closer to Jericho Turnpike.
It is capitalized when it forms part of the proper noun. Example: Syney Public Park
We had fun at Prairie Trail Park.The W in we as it is the start of a sentence and Prairie Trail Park as it is a place name/ proper noun.
In general, "county road 137" would not be capitalized unless it is part of an official title or name, such as "County Road 137."
Only if it begins a sentence, or is the name of an actual road, for example: Chorley Road.
Wilderness Road State Park was created in 1993.
Blackburn Park Road F.C. ended in 1901.
Blackburn Park Road F.C. was created in 1875.
Salamanca Island Road Park was created in 1964.
Topaz Road National Park was created in 1977.