is there a way to still open the apps i legally purchased? what if that company that wrote the softwear doesn't exists any more??
No, you don't need to re-buy programs to get them to work.
That said, some programs (including Parallels) need newer versions to be compatible with Mountain Lion. So you may need to buy a Parallels upgrade for it to work. You didn't mention the version you have but since only the latest Parallels version is compatible, you may need to pay in this specific case.
Most other apps should run OK, if they're relatively recent. You need to check with the vendors of those applications to see if they run on Mountain Lion or if they have an upgrade.
Applications bought via the Mac App Store are pretty much all compatible with Mountain Lion.