Do my parents understand it? No. They don't say anything about it typically, though. And I am very close to my parents - always have been. But even moreso in the past couple years.
They have pretty much always just let me do what I will, for the most part. I live with them, so I don't do anything that will really bother them if I can help it. Though, for the most part, they are fine with whatever I may do so long as it isn't dangerous or something.
They know I went to a furry con, but never said anything. I have a large number of stuffed animals, and they find it slightly humorous but little else (my parents, mom and dad, recently started sleeping with plush figures as well - a Grinch and Max, and still do now that Christmas is past
). And my mom got me a t-shirt with Snoopy (I love Snoopy!) and a plate that is painted with a couple of wolves on it for Valentines.
At first they [dad, anyway, mom is...well...usually oblivious
] made some jokes at the expense of it. "Tail" jokes and dropping the word "yiff" jsut to see me get flustered/stutter. But, that's the way he is. Now though? Pretty much just leaves it be completely.