How many different Parallel Universes were shown in Parallels?
How many did we get to watch Worf interact with in the episode? It's hard to determine. Do we base it on how many visiible dizziness spells Worf had, or how many changes he notices whenever he's around Geordi?
1. The cake changes
2. Picard appears at the party
3. Picard disappears from briefing
4. Data's painting moves to another wall
5. Data's painting changes visually
6. Troi's clothing & hair change
7. Worf blacks out at his quarters, & awakes at a reconfigured tactical terminal (Data is later shown with blue eyes & Worf/Troi are married)
8. Worf's uniform, & position change, & Dr. Ogawa is replaced by Dr. Crusher
See, as in #8, it's possible that some of these changes happened together, & in one quantum reality Troi's uniform (#6) & Data's painting (#5) both changed. Also, the cake changing (#1) & Picard appearing at the party (#2) could have been simultaneous.
So, the other option is to note Wor'fs spells. He only has dizziness spells once at his birthday party, once during the briefing in engineering, once in his quarters with Troi & Laforge, & then again which places him at the altered tactical terminal, & finally when they activate dead Geordi's visor. That's 5, which would mean we had seen 6 different realities that we know of
But the flaw with that is that #4 & #5 both are shown to be separate realities. The painting is moved, but the same, & then it is different, & Worf is not shown to have a dizziness spell between, that I recall
Then there's another theory that the original birthday party is actually a different quantum reality than Worf's shuttle trip, because at the end of the episode, Worf returns to his proper reality, & there is no birthday party
So, basically, depending on how you determine it, there could be as little as six, or as many as ten, but ultimately, the real answer to your actual question... How many were shown? approximately 285,000