In my experience there seems to be three types of mom's that will let a BF discipline her kids: the neglectful mom, the scared mom and the "I wanna be the good guy" mom.
The neglectful mom is just that, she doesn't care. She's more concerned with herself and keeping the BF happy. The BF may be a brutal abuser or he may be the victim of abuse at the hands of the GF and he wants to do the right thing by the kid by addressing discipline problems and trying to fill in the gaps that neglectful mom is keeping open.
The scared mom is the victim of abuse herself and the BF keeps an iron grip on the household, which includes domestic violence against mom and kids.
The "I wanna be the good guy" mom is an attentive mother with an attentive partner but she either doesn't want to be the disciplinarian or doesn't feel confident enough in herself to address issues of discipline OR she comes from a more tradtional household where mom is the attentive, emotional parents and the dad/BF is the strict one who deals with discipline.