I live in Cardiff, and tend to take a strong line on this:
parks need to be opened up to cyclists as much as possible to increase permeability. There are several parks in the city where cycling is banned by bye-laws, but this often means that law-abiding cyclists are forced onto potentially dangerous routes as a result.
Example:
Roath Park. This is a very large park, split into three. Cycling is only allowed in the southernmost part, which is sports fields surrounded by a wide path. The roads which surround the park (especially get a lot of commuter traffic (and attract plenty of speeding) at rush hour. When I lived to the East of the part of the lake with the boating lake in, I used to take my kids to nursery every morning in the trailer on my way to work at the University (then pick them up from nursery at lunchtime, returning on the same route.
This meant I had to either (i) be a good boy and cycle on very busy roads to avoid the no-cycling portions of the park, (ii) cycle on narrow footpaths with a wide-bodied trailer, or (iii) cycle through the park with them (thus giving them a slightly more scenic ride too).
Naturally, I used to break the bye-laws and go through the park (slowly, and considerately - but then I would hardly be set to break speed records with two children on tow). The paths in the no-cycling parts are mostly no narrower than the path around the southernmost part of the park. Neither are they narrower than the paths in Cardiff's largest park, Bute Park, where cycling is allowed.
Despite my lack of speed and efforts at being courteous (and despite having two children in the back), the numbers of angry comments I received from other park users beggared belief. Of course, I continued to ignore them, just like the scores of other people who commute through the park by bike (including all the secondary school kids who cycle from Roath to the high school at the northern end, and would otherwise have to break UK law by riding on the footpath or risk the car-choked roads).