“Public” parks are usually private land owned by the local council. You don’t need their prior permission to use the land, but they do retain a right to declare anyone they don’t want there to be a trespasser, and ask them to move off the land. If they did have a drone policy, that would basically be a statement of who they would consider to be a trespasser.
Lancaster City Council do have a Public Spaces Protection Order in place, see Public spaces protection orders (PSPO) - Lancaster City Council, this covers certain areas in Morecambe, Heysham and Happy Mount Park. It says nothing specifically about drones, but it does make it a criminal offence to
cause annoyance, harassment, alarm or distress to any person within the Designated Area or on land adjacent to the Designated Area or to any person living nearby
There are other legal issues anyway with causing harassment, nuisance and invasion of privacy, but the PSPO makes it all much easier for the police to deal with.
They have drawn up their PSPO maps to include beach areas, but I’ve no idea whether that’s legally valid or not.
HOWEVER … you mention following the drone code. From the maps, it looks like all of Happy Mount Park is within 150m of a “congested” area, so the drone code won’t let you fly there. Some areas of Williamson’s Park are outside the 150m limit, but not many, so it doesn’t look like me like a great place to fly if you’re keen to follow the code.