You are not sisters.
You are the Mom. She is your daughter.
She is only, 13. 7th grade? 8th grade?
She is only a kid.
How much time, do you spend with, your daughter?
What is the priority?- going out with Boyfriend. Or, spending time with your daughter and knowing how she is doing in school and what her homework is, and how her friends are and who they are?
She does not have to, like your Boyfriend.
Kids, often have gut instincts too. And you need to listen to that.
My kids, for example, have very good instincts & vibes, about people, and I listen to that. We discuss it.
Do you have discussions with your daughter? As a Mom, with her daughter? Not about only your boyfriend... but about HER life, about her feelings, about anything she needs?
Do you know... your daughter well?
As a Mom, not as a "sister" or buddy.
Your daughter, should not be dumped off at friends houses, just so you can go out with Boyfriend.
Do you even KNOW... the parents of your daughter's friends? Do you know, your daughter's friends, well?
At some point, your daughter's friends (and their parents) are going to talk about you... and how you are going out with your Boyfriend and dumping your daughter off on whatever friends, just so you can go out with Boyfriend.
Kids talk.
Kids know.
It is not pleasant to be the kid, that is getting handed off to others, just so the Mom can go out with Boyfriend all the time.
I work at a school... you would be flabbergasted... at what kids TELL, to people at school, be it other kids or other adults/school staff.
Your daughter, should not have to "compete" with your life, or your Boyfriend. And she should not be put in a position... to have to do that.
Even if you/daughter were living in your own place and not with your parents... they still probably would not support your dating this guy.
AND, if you and your daughter were, in your own place without your parents... then WHO, would watch your daughter for you? You'd leave her home alone while you go out on a date????
I think, its good you are with your parents at this point. Why? because, with your dating... it is your parents that would be keeping track of your daughter and how she is, adjusting or not, to your dating life. And at least, HOPEFULLY, your daughter has your parents to go to, with any concerns/daily issues etc., since you are not home and too distracted with your Boyfriend.
And, there is no way, that after 1.5 months of dating, that you think this guy is all perfect and "the one."
That is naive.
Of course your parents do not have to support your dating.
They see, how it is affecting your daughter.
A girl, at this age, needs consistency. They need their Mom. They do not need, to be upended. They do not need, to be forced, to like a Mom's Boyfriend and to pretend that that Boyfriend, is their male role model.
Girls this age... are still not mature. Emotionally.
You need to realize that.
And when they see their Mom, dating and doing stuff etc., then you better hope your daughter, does not REBEL nor get into trouble herself, and hope that she does not copy-cat you.