Generally speaking no, parking locations with HONK do not have a grace period of any kind.
You must pay for parking upon arrival, and extend your parking if you need more time.
Depending on what location you park at, some locations capture your license plate on entry and exit of the lot. Your payment must cover that entire time you are parked otherwise you can be mailed a ticket.
Other enforcement options can be physical ticket on your dash, booted or towed.