If you consider training in a public park you might want to check with the local authorities to make sure it is legal in you area
To answer DD's question in my area there are specific rules as to what can and can not be done in our local parks. Most of the time the police do not care and will not bother anyone unless a complaint is lodged.
I have used the parks and even had weapons training there but then I do know some of the local police and even have trained with them so they know whats happening. Most people do not mind if you use the parks but some may call to report people fighting or being a danger to others with the activities
Speaking very generally -- and with a clear understanding that there can easily be laws and local regulations that would either explicitly or effectively restrict what you can do in a park -- you're not likely to run into a problem if you're pretty clear with what you're doing. And if you do -- it's usually pretty quickly and easily cleared up. Now, if you're doing a lot of training in street clothes with little to indicate to an outsider what's up, yeah, you might find yourself the subject of confused calls to the police. Uniforms or even just t-shirts can go a long way to preventing that, as can simply being open about what you're doing. I'd avoid a lot of paramilitary looking clothes, too...
That said -- there may also be restrictions on what you can do where, within a park. For example, I live near a national battlefield park. Park regulations limit activities like playing Frisbee or catch to picnic areas. I also know of at least one occasion where a group wore out it's welcome in a picnic area by simply being too big, and monopolizing too much of the space. So that sort of thing is another point to consider.