Looking high and low for a rule...

[UmpJM]

Bainer,

I agree with BlueZebra that there is no such rule, the coach is making stuff up, and the "powers that be" are drinking kool-aid they really shouldn't ought to.

The closest thing I can find that actually addresses your question are the following excerpts from the MLBUM:

2.2 CONDUCT OF PLAYERS DURING PROGRESS OF GAME

.... During the progress of a game, all players and personnel of a club must be on the bench except when they have some duty to perform as coaches, base runners, batter, or on-deck batter. Players of the side at bat will not be permitted to fraternize with players in the bullpen either of their own or of the visiting club, but must come to their own bench even if, in their judgment, they are not likely to bat in the inning.

Only pitchers, catchers, players serving as catchers, and coaches should be permitted to stay in the bullpen during the game.

A substitute player who will enter the game at the end of a half inning or a designated hitter who will enter the game as a defensive player will be permitted to warm up in the bullpen.

...


7.2 RESTRICTIONS ON PITCHERS WARMING UP

....

A game pitcher may return to the bullpen between innings, provided doing so does not delay the game in any manner.

By my read, the players were perfectly legal in their actions in the bullpen as long as it wasn't unduly delaying the game.

Feel free to share this with "the powers that be".

JM