Looks like I was way off on my thinking on this one, even in those first 2 years before Allen emerged he scored a TD in almost all his games. Maybe you get to this number if you only look at passing TDs (there were several games where he had no passing TDs but 1+ rushing TDs) or erroneously include games where he was out injured and Peterman/Barkley/Anderson were filling in for him. So now I'm curious as to where this number actually comes from and if it is just plain wrong or misinterpreted.
Under the most generous reading:
2018: Ravens (Allen didn't start but did play)
Packers
Texans (Allen started but got knocked out of the game)
2019: Jets (season finale, Allen got the "start")
2020: Jets
2021: Jaguars
2022: None
2023: None
So that's 6, and a couple of those are a stretch as to whether they should even count. So yeah, no idea where that 14 number comes from.