Cycle routes through green spaces such as parks can and should form a vital part of cycle networks.
However a persistent problem with these kinds of routes is that they are often inadequate in width, pushing walking and cycling into conflict. The long-term solution to these problems is not to plead for good behaviour, or to ask for mutual respect, but to design infrastructure that is good enough to avoid conflict in the first place.
The paths in the Vondelpark in Amsterdam are very wide, enough to accommodate the very heavy cycle flows through the park without causing issues with people using these paths on foot.