6 months is a very short time for your DP and DD to learn to live in harmony. However much she likes him, DD may resent him telling her what to do and DP will be making loads of parenting mistakes as 1) it's fairly new to him and 2) he doesn't know your DD that well. So, yes, parenting classes would be really helpful and it's great that he's prepared to try this. Also, going together will make sure you are both 'on the same page'.
In the meantime, it can be useful to agree your household rules with your partner and the consequences of breaking a rule, then sit down as a family and explain it to your daughter.
If he wants your DD to do something, for example, the rule could be he asks once, then walks away. If DD doesn't respond there will be a consequence (that you have both previously agreed). There is no discussion, or argument, or shouting involved.
Perhaps she could have a daily 'job' and maybe have some pocket money, or extra TV time, for helping out.
Lots of praise for her efforts can also work wonders.
I believe this relationship could work BUT he has to be the adult here and if he gives up trying, or continues to try to bully your DD into doing what he wants, I personally would not be able to tolerate that childish behaviour. (Not attractive either, is it) and I would have to end it.
I do hope that this all works out for you and your DD. Good luck with the parenting classes.