I don't think Parks was ever on UK Netflix, but it was on the US one, back when you could fiddle with the DNS codes, which they've since clamped down on. It's definitely on Prime though.
Brooklyn 99 is both a great watch and light hearted enough to have on in the background. Running storylines are rather subtle so you can jump in and out of it and not be totally lost. Really easy to watch, great fun. It definitely gets better as it goes on too IMO.
Echo the above on Trailer Park Boys. It's a must if you dig Sunny. It's just horrible people being horrible and getting drunk, similar to Sunny. It starts off a bit low budget, but it gets better. Some of the acting in it is hilariously bad, which adds to the charm of the show.