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Hi. I have been looking for this for years, with no luck. I remember an episode where one of the puppets went fishing with a hoop and caught a boot. Another episode was about a babysitter the youngsters didn't like. I also remember the episode when they was afraid of the tree at night, and the thunder episode. I remember in one episode a puppet was having a bath and sang a song; "oh I love my bath, I giggle and squeal! I swim like a fish and quack like a duck. Oh how I love my bath!"
Did you ever find out what it was called?
Edited by MizLizPeppermint Park, maybe?
https://www.youtube.com/watch?v=JZwMDRKee0E
Here's the VHS cover:
https://www.amazon.com/Peppermint-Park-1-Magic-Moments/dp/B0006V8WI0/ref=sr_1_6?ie=UTF8&qid=1474829931&sr=8-6&keywords=Peppermint+park

I know for certain this wasn't Jim Henson's Animal Show or anything similar, but it was a very cute, bucolic puppet show that I used to have on VHS when I was little. As I recall, the VHS tapes had yellow cardboard slip-on covers with stripes (red on one cover, green on the other). The puppet characters were all animal people like rabbits and hedgehogs that wore clothes and lived in families, and the VH Ses alternated between live action puppet shows and illustrated pictures (this may imply that there were also tie-in storybooks, but I'm not sure since I never recollect having or seeing one of such). The plot of one of the live-action segments was about a bunny(?) going out at night and being scared by freaky looking shapes and shadows in the dark, only to be found in the morning by a friendly older puppet, who explained that what the child puppet saw were just things that looked like scary faces and figures- I most clearly I remember him explaining that one was just an old tree with knotholes that looked like a scary face. One of the storybook segments featured another bunny being afraid of thunder and lightning during a storm, and not being able to go to his older brother for help, because the brother was too busy doing homework. I also remember that the opening and ending theme was a chorus sung by the voice actors of the puppets, with a title sequence that portrayed the puppets singing in a choir together. I have fond memories of this show and I sure do miss it, so I'd love to see if I can't rediscover it somewhere!
EDIT: Solved! The series this was from was "Happy Ending Stories", adapted from a series of picture books; the adventures I recalled were "Adventure in the Dark" and "Hoppity's First Thunderstorm". Also, turns out the main character of "Adventure in the Dark" was a field mouse, not a bunny. Whoops!
Edited by WakingPrinceMatty