Firstly, nice work on getting it to this stage.
I'm wondering if your screenmode selection is the best choice for the game though. At the moment it's 8 bitplanes to draw each frame and you're a bit limited by the colour palette (Hagar and the backgrounds having to share colours etc).
I wonder what the game would look like if you switched to 32 colour mode (or maybe 64) with this setup instead:
Use 4 sprites for the parallax layer, but attached so 15 (unique) colours. Have a copperlist that repeats the sprites across the screen so you would have a 128 pixel wide background across the entire screen. This would eliminate the gaps that appear in certain places.
If you wanted to keep the scrolling parallax, you'd probably have to shift the sprite data (although on 68020+ this isn't such a problem with the CPU). Especially if it scrolls at a maximum of 1 pixel per frame. You could also use another couple of sprites (repeated) as clouds or something high above the buildings that would add another layer.
This would mean you can have the remaining 31 (or 63) colours for the game, and so your sprites could look more like the arcade. I reckon in 6 bitplane mode, it'd be hard to tell Amiga graphics from the arcade. With a nice palette selection, even 5 bitplanes would look great!
Keep a clean copy of the level background graphics in memory so you can restore backgrounds behind bobs with one blit.
I would personally forget about having the player as a sprite. If you sort all your graphics in ascending Y position (with the hotspot being the player's position on the ground), then you draw them in that order and they will always correctly obscure each other.
Well done no matter what you decide (including ignoring me completely!) BTW did you rip all the sprites yourself or take them from various versions on the spriters resource website?