If you notice your pitching staff getting heavily fatigued at the minor league levels, it usually has to do with the composition of your pitching staff. For instance, if you have starting pitchers giving you less than 5 innings a game, then you know your bullpen is overworked.
It's important to assemble a rotation with pitchers than can give you decent innings or else the entire pitching staff will see the snowball effect. The best way to see if this snowball effect is happening to you, view a few pitchers' game logs for the entire season. If they continue to enter the game at 0%, then you know that you are in trouble.
To construct a decent pitching rotation, you must make a combination of moves. You can promote and demote players, sign minor league free-agents, make trades, waiver claims, etc. You may also take advantage of tryout camp players.
Sometimes, it's very important to have some extra arms around at certain levels of the minors even if they aren't prospects. These additional arms can be placed on the inactive list. From there, the AI is smart enough to shuffle pitchers to and from the inactive list to preserve pitchers' arms.
These tryout camp players are essentially filler -- they exist to eat innings and prevent your true prospects from being overworked. The lower the minor league level, the more pitchers you need on your inactive list. If you have a properly constructed pitching staff at Low-A, you need 4 or 5 relievers on the inactive list. At AAA, you can probably get by with 1 or 2.