Should there be an apostrophe in the sign "residents parking". A quick Google search suggests there shouldn't. But why not? Is "residents" an adjective?
I would recommend this: "Residents' Parking" because the intent is to provide clarity. Who is allowed to park there? My interpretation of those words means that this area is where (only) residents may park, and there is more than one resident, so it should be plural possessive.
If it were a single parking space meant for one residential unit, then "Resident's Parking" would be appropriate, but not typical, and not as clear as "Parking for Unit 1" or "Reserved for Resident" or "J. Smith's Parking [spot]."
Without any apostrophe at all, the words "Residents Parking" is like a caption on a photo of a parking lot showing cars in the act of parking, like enlightening visitors to the zoo: "Gazelles Grazing."