milauskas,
I had a similar task and wanted to use the mini calendar for the function. Even though I was able to create a filter for the minicalendar to display only events that were in a specific category (as your example, “Parents” category on a Parents page (matching slugs)), I still had an issue with the list below the calendar displaying events from all categories and since I am on time constraints I decided to go the route of simply listing the events in a list.
Here’s what I did:
http://pastebin.com/Zg1BBLNT
You’ll see this isn’t completely optimized and doesn’t address using the minicalendar, but it’s definitely a step in the right direction and provides a solution for you to build from.
I’ve stripped it down to basics — I have more details to display but wanted to share for the sake of example.
Another option is to register a sidebar for each individual page and place a new instance of the calendar widget with manually-set category filters.