This is easily the most fleshed out game I've seen so far. You really went above and beyond and it shows. The parallax backgrounds in the cave level in particular really impressed me. The artist did a wonderful job
You clearly have some handle on game development on this point, so I've got a few additional pieces of feedback to help improve your platformer design if you keep going in the platformer genre:
1. Holding down the jump button shouldn't auto jump. In godot, instead of using Input.is_action_pressed you should be using Input.is_action_just_pressed so the player has to press the button each individual time they want to jump. Auto jumping feels unpolished/clunky. It was evident to me in the bat chase sequence that you clearly used autojumping when testing, because that was the only reasonable way to clear that sequence (bug btw: you can jump over the scroll and never spawn the bats). Don't rely on jank in your designs.
2. Coyote Time: This is a mechanic that seems counter intuitive at first, but is used in pretty much every well liked 2d platformer in recent history. If the player has just walked off the ledge and pushes jump, you should still allow them to jump for the next few frames. This creates the illusion that jumps are "fair" to the player, because the action they meant to take is translated to their on screen movement, even if you as the designer know they 100% just cheated
3. If you're going to have a sword, make it one swing per button push. Having the player swing up and down makes it feel sluggish. It's almost like I'm telling the character to swing their sword rather than swinging the sword myself. There is a disconnect between what I'm trying to do and what I'm actually doing
4. The bats move too fast and their pathing is random/unpredictable on the first few attempts of the cave sequence. Enemies need to move in a way that is predictable to the player so they can anticipate and react the first time. They shouldn't have to die and restart to beat your game (though they probably will anyway)
5. Be more generous with checkpoints. Restarting the entire cave level when I failed on the bat sequence was what encouraged me to realize I could just hop the bat spawner, and after just barely hitting a complex jump in the bar sequence (more on that later) dying to a knight enemy caused me to end my playthrough there rather than restart the entire sequence
6. In the opening to the bar level, when you descend into the cellar you pass by a row of barrels with no collision on them only to be stopped by a rack of barrels in front of you with collision on them. There is no visual distinction on why one pair is in the background while the other is in the foreground. In general, you need to make the foreground brighter and the background darker
7. Later on in the bar level, there is a tight sequence where you are supposed to drop off a ledge, wall jump off the side, double jump to the other side, wall jump again and reach the top. The issue is there isn't enough room for the initial drop, meaning it feels more like luck than skill if you wall jump off the left platform at the correct position, way too high, or just fall off entirely and have to repeat the entire sequence. The platforms that hang down for that sequence need to be elongated
8. Double jump should not be tied to a second button. Just have me press jump twice, don't throw extra buttons into the mix
Hopefully all of this will be helpful to you in some way. Congrats on all your hard work! I can't wait to see what's next for you