Hi putri, some ulama differ in their opinion on getting married while pregnancy. For us, we follow mazhab syafi'e. Hope this link clears your doubt http://ustazlove.com/blog/2011/04/soal-jawab-bernikah-dengan-wanita-mengandung/#
From what i knw, based on a hadith by Rasulullah S.A.W. , the child cannot have the father's name as the child is considered to have no relation to the father. Even if you marry the biological father, the relationship they will have is tht of a stepfather n stepchild.
I think it is best to let your parents knw. I know it must b very difficult to do so but you have to let them knw. Eventually they'll find out. you r already in your 13th week. They will knw coz you'll b delivering in your 37th - 40th week. Ppl make mistakes n the best thing to do is learn frm it n move on. Parents will get angry, but for awhile becoz they love their child more.
You can choose to nikah 1st at ROMM. Since you r aready pregnant, you need not go for the marriage prep course. They will have another course for you to go thru after you deliver. In the meantime, ROMM will hold on to your marriage cert till you complete the course.
I hope this helps. N i hope Allah gives you the strength n courage for you to speak to your parents about it.
We r always here to hear you out.