I recently watched Spring, Summer, Fall, Winter... and Spring and I loved it. However I detected certain symbolism in the film that I couldn't decipher to myself. In the Summer, the monk and his apprentice have a rooster with them, in the Fall the master brings a cat and there is a snake involved in the winter if I'm not mistaken (it's been couple of months since I've seen the film so my memory is a bit fuzzy). I'm assuming that those animals have a certain symbolic meaning and/or are connected to the Buddhist iconography so if anyone has any deeper knowledge about this it would be much appreciated.
This is a fair and interesting question. Spoilers for the film follow.
In Buddhism, a rooster (or really just a bird in general) is one of the three poisons of life, along with a pig and a snake, and the bird specifically represents greed, and they are depicted at the center of The Wheel of Life. In the case of the film, it more particularly symbolizes lust - the lust the apprentice has for the young convalescing woman. Refer to this article for more. At the end of the summer section, when the apprentice escapes with the rooster and the monk's statue, he is both burdened by lust/greed (as symbolized by the rooster) but also by the monk's teachings, adding a touch of irony to the situation.
During the fall section (my personal favorite), the monk brings back a cat from a supplies run and the cat leaves when the police apprehend the apprentice. According to this source, in Buddhism, "the body of the cat is the temporary resting place of the soul of very spiritual people". When the monk brings back the cat (before he has gained information about the former apprentice), it demonstrates that at his advanced age, he has, in his solitude, achieved some form of inner peace and foreshadows his death. When the cat leaves with the apprentice, it perhaps symbolizes the monk's dying wish - that the apprentice finds inner peace as well. Check out this source for more.
In the winter section, I do also recall there being a snake, though where in the section it appears I do not remember either (it has also been some time since I've watched the film). In The Wheel of Life, the snake is another one of the evils, along with the rooster/bird - it represents hatred/anger. In the film, the apprentice returns to discover that his master has died, so his anger and hatred - perhaps directed at himself for not having followed the monk's teachings during his lifetime - is not unreasonable. The fish, frog, and snake he tortured in the spring section return (in Buddhism, goldfish symbolize happiness/freedom and frogs are considered symbols of good luck; it is perhaps ironic that these two were tortured along with a snake).
I hope this helps! Though I am not close to finishing Kim's filmography, animals have appeared in other films of his that I've watched - The Isle and The Net involve "fishing resorts" (and thus fish), Address Unknown features a protagonist who owns a dog, and in Pietà, a rooster and a fish appear. Writing this answer makes me want to rewatch Spring, Summer, Fall, Winter... and Spring!