No, you are again going back to restricting a particular group. That's NOT what is happening. They are enforcing the same parking regs for everyone, everyone is allowed to park. Now some decide they don't like how parking normally works, so they should have the right to park in a different way. That's fine, but the state does not have to have some sort of strong basis for not wanting to redefine what "parking" means.
They do have a rational basis to enforce parking in a uniform manner... it provides more parking capability.
An interesting point was made recently regarding parking in my curb breakdown where my driveway meets the street...
My son, a SDPD cop, parked his truck for a brief period to visit his grandma in the area where the curb is absent to enable egress from my driveway... He got a ticket from the wonderful parking enforcement officer... for illegal parking...
He went and argued the issue but they were relentless... pointing out that in my town parking is enabled with in so many (18 inches) of an otherwise unrestricted by painted surface (red/green/yellow etc.) curb... and where there is no adjacent curb formations... like in the dirt off the side of a road where there is no curbs.. so long as you are well clear of the road... we have areas like that still..

and some other conditions as well...
The logic of getting a parking ticket where he was made no sense.... but they argued that all people are restricted in that way... that your parents don't mind is not the issue... what would they say if the neighbor parked there... and if you can why can't he? He lost his case.... ()
Parking means the leaving of a car unattended... the stopping but being in attendance in a restricted or unrestricted area for a period of time may also constitute parking.. like at an airport... So, when a case is brought to argue the definition of parking we may find that there are other conditions that meet that test...