"An originalist understanding of the Constitution would consider parking mandates a taking of property in violation of the Fifth Amendment's Takings Clause protections, incorporated against state and local governments by the 14th Amendment. Congress would therefore be empowered to pass legislation to protect individuals' rights from these parking mandates."
There is no taking, because the restriction is already voluntary, and in consideration of mutual benefit of like-zoned neighboring property being similarly restricted.
It's obviously evolved into something else, especially as seen by the public, but that philosophy remains the legal foundation for it.
Even if zoning changes require approval, only the property owner can determine the original zone, and only he can request changes.
Buying zoned property, knowing the extant restrictions, is essentially opting-in "voluntarily". It's the same reasoning for why "grandfather" exemptions exist (no original consent).
What this federal argument is actually for, possibly without realizing it, is a method/mechanism for contractural/ municipal/ jurisdictional succession and/or nullification.
Those are the terms that need to be thought of in context of this discussion.
I have no idea where you live, but I assure you that in many places the authority have jurisdiction can in fact just change the zoning of property you already own.
It seems internally inconsistent to apply originalism to parking at all in the sense that there is no original thought on cars at the time of the drafting of the constitution simply because cars had not been invented yet.
See Bruen where a lack of laws regulating guns indicates public carry is a right rooted in historical tradition vs. Dobbs where a lack of laws regulating abortion before quickening does not. [1]
The conservative legal movement has turned constitutional interpretation into a shell game where choice of framing and selective use of historical examples can yield whatever result the justices already want.
All of that said, the argument is that parking minimums constitute a restriction on your property rights for public use without compensation and a similar argument could be made with things that did exist at the time such as carriages. Again see Bruen where Thomas "refutes" a weapon ban by arguing handguns are more analogous to daggers than to lances.
[1] https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4187143