Hey,
is there a way to execute a command like /lp user ... parent add player with a command block?
Yes, exactly this question was here several months ago and it was closed with: "Command blocks are - to my knowledge - not supported by plugins or spigot in general." (#1721)
I have the same problem but I performed some tests and here is my results.
I have command block with this command:
/pex user Adam group set group-a
There is not "@p" but specific username ("Adam").
Spigot + PEX: it works
PaperMC + PEX: it works
PaperMC + LuckPerms: does not work (with /lp user Adam parent add group-a and /luckperms:lp user Adam parent add group-a), nothing happens.
So I think it is not problem of Spigot/PaperMC but LuckPerms.

Assuming your tests are accurate, it does indeed seem to be a LP issue.

Hopefully I'll get some time to look into it soon - if anyone else wants to have a try in the meantime that would of course be appreciated.

Command blocks w/ LP work fine on my server (Tuinity 1.15.2 but also worked when I used Paper)
The only thing I have that could maybe be making it work is the fact that I have CommandHook installed

I've tested using the above setup and it seems to work fine. Closing for now.