You remind me of an anecdote about an American working in Asia. They were constantly frustrated with communication difficulties. One day, they felt they finally were on the same page with a coworker and they said "We are thinking along parallel lines." to express this. Coworker agrees.
Later, the problems came back. They referred back to that conversation. Their Asian coworker says "Yes. Parallel lines never meet."
Gotta love the straight German way.
It is usually not the best explanation and it actively causes all kinds of problems. I like to give push back against this common assumption wherever possible.
The Pan Am smile: https://www.urbandictionary.com/define.php?term=Pan%20Am%20s...
Is it? The description on that page is just about having multiple conversations at once, not having them all start in the same place.
Also, starting in one place and going different directions is basically the opposite of parallel lines, so this makes zero sense as a justification of the punchline.