There are no details here that give us clues.
I don't know how long the two of you dated before becoming engaged. If it was quick, that might be cause for concern. He may not have been in recovery long. As I understand it, those in recovery ( in AA, anyway) are strongly discouraged from entering into new relationships when they are newly sober. This could be a concern for them, even if they aren't aware of the guidelines in AA.
Renting as opposed to buying, that bothers them? I don't know why, other than maybe they think he may move and take you with him. Is his job stable? For how long? You might tell your parents that buying doesn't always turn out better than renting. Your parents may have fared okay, but many people were encouraged to buy houses when they couldn't actually afford them, and ended up worse than before.
An acquaintance of mine married a man in recovery, but they had a long courtship. She insisted he be sober for a specific length of time before they began planning a wedding.