Yes, you can uninstall Vegas Pro 13 without it affecting Vegas Pro 18. Upgrades to a new version are based on your having a serial number for a previous version (but NB that a serial number can be used only once for an upgrade). The actual installation of a new version on a computer does not involve that installation looking for a previous version of Vegas Pro on that computer.
@Vernon-Hodgins An upgrade to VEGAS Pro 18 is a totally different version, so you can safely uninstall VEGAS Pro 13 if you so desire. However, space is cheap nowadays so it may be a good idea to leave it installed. You may need it sometime in the future. You never know.
Some shared plug-in can be removed by uninstalling, but this occurrence is usually with a third-party uninstaller/cleaner. Regardless, multiple versions of VP can co-exist w/o issue. IMO, if it ain't broke, don't fix it, and it does not use very much drive space.