Doesn't matter how strong the case for paying minimum wage is, Congress can pass a law - the very law proposed in 2016 - exempting Minor league players from the minimum wage laws. That law would override current federal law, pre-empt state wage laws, and end the suit. (Once MLB has an exemption, it can even reduce wages for MILB players.)
(The law could be written to "clarify" MLB's position that minor leaguers were always intended to be exempt.)
I'm surprised that mayors from the towns affected aren't complaining - the law would directly impact and adversely effect their towns. Increasing pay by even a few hundred dollars a month would help all a town's businesses as the players would turn around and spend their largesse at local businesses.
There is something that parents/players can do: call your local representative and educate them. Most don't get deep in the weeds to understand a bill; in this case your rep would need to understand the difference between MLB, MILB, who pays what costs, who owes what obligations and who would be hurt (local towns) and who would benefit (large corporate absentee owners). If your local representative has an MILB franchise in the district, it should be easier to educate him/her.
So, make the calls!