I wanted a progressive education because I wanted my kids to enjoy learning and to appreciate different perspectives on a problem. Facts become obsolete, and blind memorization of them, or learning "the answer", won't stand you in good stead in the long run. The process of exploring a problem is more important than "the answer". I feel that kids learn better by "doing" rather than by sitting in class taking notes while a teacher lectures or sitting in class reading your history text. (This is just an example of how I was taught history, and explains why I was completely turned off to the topic). At Park, for example, students might engage in debates in history. They might celebrate "Modern Languages Day" by forming small groups to write and act out skits in which the audience is expected to interact. An English project in middle school will involve crafting a "speech" on a controversial topic in which they have to acknowledge multiple sides of the matter. In lower school they might form small companies and enter into a classroom competition to see who can design the strongest bridge - while mastering business and design skills. Kids really become engaged in the material.
I will say that the political hues at both Park and Friends are pretty liberal. This was perfectly fine with us as we're that way ourselves. I do know at Park they do try to value all opinions; they had, for example, a "Young Republicans" club in the upper school at one point (which my older son joined in order to hone his debating skills). My older son also especially enjoyed the "Student-Faculty Forum" in which a controversial topic was debated each week between students and faculty. An education is so much more than just having as many facts as possible crammed into your brain...
For example, I appreciated the number of openly gay teachers at Park. No need to hide in the closet, especially now with a gay head of school. It's not a place where homophobia, racism, or religious bigotry would be tolerated. (And lest you think that's true anywhere, I should tell you about some of the shenanigans I've seen team members from other schools - or their parents - pull over the years...).
The days are pretty structured in the lower and middle schools. I happened to like the relatively loose structure of the school day in the Upper School. Kids who were on a team could still participate in their sport, but weren't precluded from participating in other activities, as many met during the "free" blocks between classes. It meant that homework was never assigned one day to be due the next - great if your kid had a music lesson after an away game with little time left for homework. It meant that my kids have been able to pursue a wide array of activities, further enhancing their personal growth.
As far as tidiness goes, I rank that as very low on the list of things I'd be concerned about. My younger son's last school was tidy, and anything but chaotic, but he was bored out of his mind with the style of teaching. He did not enjoy school at all. Whereas now he'll come home and tell me about the interesting things they did in some class.
What type of families would fit in... I would like to say "most or all", because the school does foster an atmosphere of tolerance and is aggressively seeking to increase its diversity. I think anyone who comes to the school with no fixed agenda and an open mind would fit in.
Ultimately, I think both Park and Friends are pretty similar with respect to the sorts of families they attract and the values they hold. That was important to me, as I wanted our values to be shared by the families of my children's classmates. Although they aren't as incredibly competitive (at least in today's economy) to get into as I've heard some of the schools in NY and DC are, they're still competitive, and I'd apply to both and see what happens.
On the other hand, if a generally liberal social or political agenda is going to be irksome to you, there are other schools you might prefer. Gilman (possibly/probably harder to get into), Boy's Latin, Calvert, St. Paul's, McDonough are all more conservative. (The first two are all-male, the last three are either coed or have boy's and girl's divisions). If you want something even more liberal or green than Park (yes, it's possible!), check into Waldorf (also more structured). (Waldorf is an especially good option for younger kids - elementary or even middle school. Waldorf will also be very unlike any other school you've ever visited, and you may find it an especially intriguing option).