Parallels Desktop 14 supports Mojave, which is what I have. I expect it should work without having to upgrade to Parallels 15. https://www.parallels.com/blogs/parallels-desktop-14-system-requirements/
I just bought parallels less than two months ago. I don't think I should have to upgrade to a new version of Parallels, especially as it was working with the MacOS fine up until very recently. I am on the same Mac OS.