They only appear to be related because they're both based around fairy tales, but it's entirely superficial. The series aren't related at all, and even have entirely different themes:
For example, on The 10th Kingdom, Little Red Riding Hood was one of the queens in the fairy-tale world. In Once Upon a Time, Red is about as far from a Queen as you can get.
Your guess about the parallel fairy-tale world - called the Enchanted Forest - is partially correct. Every episode has flashbacks back to the Enchanted Forest, where it shows the past of that episode's focus character. However, this we don't learn until the second season:
Their world wasn't destroyed. It still exists, just decimated. The Curse only brought over a small selection of people to our world - the ones Regina wanted to punish. There were other countries outside the Enchanted Forest that the Curse didn't go after with nearly as much vengeance.
The 10th Kingdom was a comedy, while Once Upon a Time deals with a whole lot more. For one example from the first season, Emma gave up Henry for adoption and is now trying to get back into his life. Later the series touches on abandonment, redemption, and in the most recent episode (3x11) there's even
a noble murder/suicide.
Aside, there's also a spinoff series, Once Upon a Time in Wonderland, that's set entirely in Wonderland, a third parallel world. It's... hard to describe. There is more comedy than Once Upon a Time, but mostly due to Wonderland itself being weird and Alice just taking it in stride:
Alice: I have a plan, Knave.
Knave: And a new outfit. Where'd ya get that?
Alice: The Clothes Horse came by.
Knave: 'Course it did.
In terms of how "dark" a series the spinoff is, I'd say it's around the same as Once Upon a Time.