챙기다 has several meanings but most of all it means to "take care of","pay attention to","pick up and collect". Here it means "tend to someone's birthday i.e. not forgetting their big days and devote one's time and energy to it. 편이 does not mean comfortable... it is a noun here. 편 means "side, tendency"... 편이 아니라 means "one does not have tendency to". to link it all up, it will roughly say "(s)he doesn't tend to pay much attention to birthday event."