My now-husband and I were in this position several years ago. I sold my home and moved into his. I paid him rent. In our case, it was just easier than trying to sort out whose equity was whose - he had a much bigger stake in his home than I did or would. We figured we wouldn't be there forever so it was worthwhile to keep it simple. It didn't bother me in the least to go "backwards" from owning to "renting". If it bothers her, that's something you'll need to work through. If it bothers her parents, they can mind their own business. (You can't control what other people think so, for me anyway, what her parents think wouldn't even be a variable in the equation.)
When we moved a year or so later, we bought a home together with both names on note and deed, each paying 50% of the mortgage. That way if we split, there was no arguing about the division of equity - 50/50. But this wouldn't have worked as well with the home we lived in originally - he already had quite a bit of equity in it and my "share" would have been so small, it just wasn't worth the drama of dealing with it if we split up.
If you plan to live there for a long time and get married somewhere in there, she might have a 50% stake in the equity anyway, depending on your state. If not, you could do a sort of a prenup in which a certain percent of equity in the house is hers, if you split later.
We're married now and - I think maybe out of habit - we got married a little later in life - we still keep finances separate. We have a joint checking account but each deposit X into it every month, then pay mortgage, utilities, etc., from that account.