No. You definitely can not open a newer project with an older version of the program.
There is no backward compatibility in Premiere Elements (any version) in this regard.
And, if you are trying to open a 12 project in 13, make sure to be working from a copy of the 12 project as a safety net. If the 12 project is opened in 13 and edited in 13, you cannot open the 13 edited project in 12.
Please consider, and give more details if you need supplemental information on the above.
Best wishes for the new year.