I was an unwed mother when I had my baby girl almost a year ago. The father wants to be a part time daddy when it is convenient for him. He did not want to sign paternity in the hospital so therefore he didn't go on her birth certificate. Does this mean that in the state of Pennsylvania he has already relinquished his pateral rights to her? And if that is the case can my husband adopt her without having to contact the birth father?

No it just means he is not listed on the birth certificate, being listed does not give or take away any parent right. ( expect now he will have to take DNA test to prove he is the father if you so require it)

For your husband to adopt he will have to formally sign legal papers giving up his rights.

If he has never claimed or been proven to be the father then he has no paternal rights. However, if your husband were to petition the court to adopt your daughter, the judge would no doubt require you to contact the birth father and give him a chance to consent or object. That would mean either him voluntarily admitting to being the father or submitting to a DNA test to prove it.

I was an unwed mother when I had my baby girl almost a year ago. The father wants to be a part time daddy when it is convenient for him. He did not want to sign paternity in the hospital so therefore he didn't go on her birth certificate. Does this mean that in the state of Pennsylvania he has already relinquished his pateral rights to her? And if that is the case can my husband adopt her without having to contact the birth father?
From what I have heard if the father is not on the birth certificate your new husband is entitled to be put on the because as the bio father.

I was an unwed mother when I had my baby girl almost a year ago. The father wants to be a part time daddy when it is convenient for him. He did not want to sign paternity in the hospital so therefore he didn't go on her birth certificate. Does this mean that in the state of Pennsylvania he has already relinquished his pateral rights to her? And if that is the case can my husband adopt her without having to contact the birth father?
Get ahold of your legal aid office for your county and they should be able to help you