What operating system is your old Mac running? Scrivener 3 will work with macOS 10.12 or above, so if your old Mac can run Sierra, you’re good to download Scrivener 3 on that one.
If it’s too old a model to run Sierra though, then yes, you can work with your projects between v.2 and v.3. However, while Scrivener 3 can convert v.2 projects to its updated file format upon opening, Scrivener 2 can’t support Scrivener 3’s file format. This means that you’ll have to use the File > Export > as Scrivener 2 Project… menu command when saving your project in v.3 if you plan to then open it in v.2.