Blackadder (you can safely skip series 1 and go straight to 2-4), Fleabag and The Marvelous Mrs. Maisel. Wait, how old is your daughter?
You could try searching out Green Wing. I don’t know if it’s available in US markets. It’s some way further along the spectrum from Parks and Recreation, venturing well into the surreal.
Talking about sketch shows, Smack the Pony is hilarious.
For sitcoms, try ’Allo, 'Allo! It’s kind of like the next step beyond Hogan’s Heroes.
A third vote for Detectorists.
Flight of the Conchords.
I’m surprised you’ve not listed The Office, either original or the US versions.
Possibly hard to track down, depending where you are, but look for:
Father Ted
The Windsors (not to be confused with The Crown), and an earlier show with many links - Star Stories.
*W1A *- set in the BBC.
It’s Always Sunny in Philadelphia
I think Wilfred would qualify as ‘quirky.’
There is the original Australian series Episode 1 and a US version with Elijah Wood and Fiona Gubelmann Explaining Wilfred.
I have to assume you’ve seen it, but if not, Scrubs. The reruns must be on a dozen times a day so it’s easy to find and due to the style of narration, you can, for the most part, watch any random episode and get up to speed pretty quickly (IOW, if you want to try it, just watch the next episode you see airing).
You could even look at the earlier series that spawned it, 2012, a mockumentary about the London Olympics.
“That Mitchell and Webb look” is a sketch series but excellent, and “peep show” has the same two people in a sit-com.
“the IT crowd” is excellent
“the day today” is a spoof news show from the early 90’s, hilarious, surreal and presient. the follow-on “brass eye” is Chris Morris at his best.
You might want to search out the Alan Partridge saga as well. He starts as a sports reporter in the above “the day today”, spins off into a chat show with “knowing me, knowing you, with Alan Partridge”, then in mid-career hiatus with “I’m Alan Partridge” series 1 and 2. There are others beyond that but they are a good place to start.
Also “Toast” is great, a struggling actor played by Matt Berry (star of the aforementioned and also recommened “what we do in the shadows”), Any show that has names such Sooki Houseboat, Dinky Frinkbuster and Clem Fandango has got to be good.
The Good Place, for sure.
You must begin with Season 1, Episode 1 and proceed in an orderly fashion.
New Girl. How do you not love a series that once persuaded PRINCE to guest star and make pancakes?
Wonderfalls would fit on this list.
Trial and Error if you liked the Office.
There’s also “Monk,” which is almost as quirky and entertaining as “Psych”
Letterkenny was going to be my suggestion, but you beat me to it. In keeping with the Canadian theme, also check out Trailer Park Boys.
Better Off Ted is absolutely one of my favorites. I discovered it on a flight, of all things, looking for something to watch that wasn’t the same old boring shit. I highly, highly recommend it. I was so disappointed it only lasted two seasons (in fact, I may have discovered it right when it was cancelled. But it’s all gold, as far as I’m concerned.)
Oh, and I second Peep Show as well.
Weeds i found to be one of the best shows ever. One thing about it: Every episode ended with a real cliffhanging bang. It was very hard to stop watching.
I loved Weeds in the beginning but grew weary of Nancy’s super asshole behavior.
May I suggest Grace and Frankie? It is about family and getting old but Jane Fonda and Lily Tomlin (plus every actor and actress over 50) know how to do irreverent. Also, it’s pilot is about how their business partner husbands announce that they have been lovers for 40 odd years and now want a divorce so they can marry each other. It gets even better from t here.
The Larry Sanders Show, starring Garry Shandling, Rip Torn, and Jeffrey Tambor: a take-down of late night talk shows.
If you like it dark: Six Feet Under on HBO.
Futureman on hulu is also damn good
If you can track it down, the '80s Canadian sketch show Bizarre, with John Byner and the late Bob Einstein. It’s wonderfully non-PC.
I just finished watching an episode of Night Court. I love Bob and June Wheeler!