My area of knowledge:
First, yes, its technically illegal to just go plant a tree on your own in a city park; however, there are plenty of ways to get what you want done anyway.
Doing it on your own, in a natural setting you might well get away with; but sticking it in the sod or landscaped bed, it would likely be mowed over or removed.
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So how to get this done.
Depends on what you want.
The City has a memorial tree program where they will plant a big (caliper) tree in a park, if you pay for it, and you can get a tiny plaque in front of the tree to honour someone. They charge you a fair bundle for this though.
http://www.toronto.ca/parks/commemorative_tree_bench.htm
Second, you can just get the same thing (a new tree added to your park) by asking, no plaque, no fee to you. Parks has a set amount of money for trees each year and another pool for 'minor improvements'. A big chunk of this just goes to little projects people ask for. See a new swing set? Could be that it just needed replacing, but there's a 50/50 chance that it got replaced cause someone called the councillor or the parks supervisor for the area.
If you would like 1 or more new trees in your local park, consider asking your councillor's office; or by-pass them and phone 311 to find the name of your local parks supervisor, and phone them directly.
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Another option if you want to get your hands dirty is to work with a group like Friends of the Don East,
www.fode.ca They run plantings every spring and fall, for the public and school groups, mostly in Toronto's valley parks. Obviously you just plant what they're already planting; but I know they've put in a unique or larger tree before if someone made a donation etc.
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Finally, on species
This depends on your location, and what you want to see, and how soon.
Examples:
Silver Maple requires full sun
Sugar Maple prefers being planted in the mostly shade (part-sun is OK)
Silver Maple and Cedar like normal to wetter sites
Red Pine prefers sandy and dry
IF your near a road, or pathway, you want something road-salt tolerant (Silver Maple is good) (Sugar not so much)
If you want the tree to grow quick, to be impressive; something like an Trembling Aspen or Balsam Poplar would be good; they reach full height (10 storeys plus) in 15-18 years; however they also die by 30.
IF you want something that will grow slowly, but assuming it survives its first couple of years, will still be growing when your children are old; Oak Trees tend to live between 80-120 years) and Cedar can live for centuries.
Cedar Trees though are very vulnerable to dog pee! (turns them black and they die, when they're young)
Native trees are still not easy to get at a local nursery, so if you decided not to go through the City, I would suggest seeing if you can get FODE or one of the other local groups to just add a tree to an order for you; or you could make a special trip to a nursery.
Evergreen at the Brickworks is supposed to have their native plant nursery up and running this year, not sure if they do yet; otherwise, you probably have to go out of town to a place like Acorus.
http://www.ecologyart.com
or
Native Plants
http://www.nativeplants.ca/
Hope that helps.
(Helps to know some old school Amish lumberjacks/carpenters who work for very little other than food on the table and a place to sleep.