To review your statement:
"I was doing Chuck boss earlier and had my Togekiss out against his Mienshao. I missed Air Slash and he used Bounce, and when it hit Togekiss he got paralyzed."
His Mienshao got paralyzed by attacking your Togekiss with Bounce?
"Next turn Mienshao used Bounce again, so I used Heal Bell. It didn't heal the paralysis. I had to use Heal Bell 4 times before I could remove the paralysis."
You tried to heal his paralyze?
"Does Heal Bell "miss" when there is no pokemon in front, when it's supposed to work 100%? I checked the not working moves list and it's not there, is it working as intended?"
Heal Bell is healing the user and your teams status conditions not the enemies status condition.
Kind regards.
Edit: Your wording was a bit confusing. Got it. I'll look for an answer right now.