Our house has a finished basement too. It has bedroom, bathroom, laundry and a kitchenette. My mom lives down there, since she is aging and I take care of her. I think basements in the US are pretty common, although in my state they aren't since the ground is too hard usually to dig that deep (Arizona). So I'm kinda lucky.
I think leaving the nest, especially if you live in the US is a normal part of life. Your parents probably just want to see you move forward with your life and be independent. Although if you are getting ready to get married, it seems strange they would want you to try and rent somewhere while you could be saving up money for a down payment on a house or for your own place.
Have you sat down with your parents and asked them what their expectations are? Maybe what sort of future they hope to see for you? When I was getting ready to get married, I wanted to move out since I had already spent 4 years on my own in another state for college. My dad was insistent that paying rent somewhere was a waste of money, and that I should be saving for a house. Then he passed away right before my wedding, so my husband and I couldn't live at home even if we wanted to. I also think that your future spouse should be present too in that conversation. It might help to clear the air and give you some direction.