The parking sign is probably a contractual offer, which can be accepted by conduct
As answered at this question, the parking sign asserts itself as an offer and even sets out the manner of acceptance: conduct by parking.1
A menu, like other prices in a store, is usually not put forward as an offer that a customer can unilaterally accept in order to bind the proprietor to perform
The owner who has posted prices on a menu may be committed to the posted terms if the menu likewise holds itself out to be an offer. However, I expect it would be rare that a proprietor would make an offer via their menu such that a customer could bind them to produce an item by the customer's acceptance and corresponding promise to pay. A restaurant is occasionally out of stock of an item, which would leave it at risk of breach if the menu were in fact an open offer that could be accepted by the customer.
See Pharmaceutical Society of Great Britain v Boots Cash Chemists (Southern) Ltd.  EWCA Civ 6 (05 February 1953). There were several sets of reasons in agreement, but I adopt the crux of the reasoning from Somerville L.J., replacing inline some terminology to apply it to your menu question:
Is it to be regarded as an offer which is completed and both sides bound when [the customer tells the server that they accept item X from the menu], or is it to be regarded as a more organised way of doing what is done already in many types of shops — and a bookseller is perhaps the best example - namely, enabling customers to have free access to what is in the shop to look at the different articles and then, ultimately, having got the ones which they wish to buy, coming up to the assistant and saying "I want this"? The [server] in 999 times out of 1,000 says "That is all right", and the [food is served, the money passes,] and the transaction is completed. I agree entirely with what the Lord Chief Justice says and the reasons he gives for his conclusion that in the case of the ordinary [restaurant menu], although [food is] displayed and it is intended that customers should go and choose what they want, the contract is not completed until, the customer having indicated the [dishes] which he [wants], the [restauranteur] or someone on his behalf accepts that offer. Then the contract is completed.
So typically, until the customer places their order and the restaurant accepts it, there is no contract.
Bait-and-switch pricing, double ticketing, etc. may be prohibited by consumer protection regimes
But regarding your separate concern that a price on the menu may later be disavowed, another answer explains:
Where an invitation to treat also constitutes an advertisement or posted price in the context of consumer protection regimes, it can take on additional significance.
What constitutes an advertisement or posted price or other significant communication in consumer protection regimes is defined by statute and has only coincidental overlap, if any, with the concept of invitation to treat.
For example, in Canada, the Competition Act prohibits a seller from charging the higher price between two or more prices clearly expressed, at the time the product is supplied, on the product, its wrapper, its container, on an in-store or other point-of-purchase display, or advertisement.
1. A commenter suggests "that the manner of acceptance is paying, not parking; and that it "would be like taking your biscuit to the register." However, these are not accurate statements of the law. Where the offeror, as master of the offer, stipulates that acceptance can be by specified conduct, doing that conduct constitutes acceptance of the offer. This is well-supported through citations to case law at the linked duplicate. The situation of the parking lot with a sign is the opposite of bringing a biscuit to the register. Posted prices for items in a store are typically considered invitations to treat, so are not subject to unilateral acceptance by a customer. This is the opposite of the case in a parking lot where a sign is posted saying that parking will constitute acceptance of the contract. Certainly, questions might arise about whether the car owner in fact "parked." If a driver places their vehicle on the lot, turns it off, gets out of their vehicle for a moment, reconsiders, and then gets back in their vehicle and drives away, I would not expect this to be considered an act of "parking" that objectively communicates acceptance of the offer on the sign. On the other hand, leaving the vehicle on the lot and walking away probably will be considered an act of parking that objectively communicates acceptance of the offer on the sign. The Federal Court of Appeal said in Imperial Parking Ltd. v. Canada, 2000 CanLII 15612 (FCA): "the unequivocal conduct which constitutes acceptance of the appellant's offer to provide a parking space occurs when the driver leaves the lot after parking his or her vehicle." But these are simply questions of fact, and the ultimate focus of the inquiry remains whether the actions of the driver were unambiguously an act of "parking" such that they were objectively manifesting acceptance of the offer. It never turns into a question of payment instead of a question of parking.