Can Adobe Premiere Elements 2020 use projects created by Adobe Premiere Elements 2013?

Maybe, but not probably.  It seems that as features changed, the newer versions may not be able to handle the features from projects built in older versions.  Nor does Adobe promote "backward computability".   

There have been posts here on this forum where opening old projects did work.  It appears that usually they were simple projects.  

You can install a free trial and try it.  Be sure you are trying it on a copy of the project because the new version may cause damage to the old project.   

The other thing you can do is use the actual output video of the old project as source media for a new project.