There is no legality in place for individuals wanting to become paralegals if they have a felony in their background.
However, it is essential to understand that many training programs will not accept individuals with felonies and may require a third-party background check before admissions.
Additionally, having a felony on your record may also prevent you from getting hired at a desirable law firm.
Before applying to any program or school, find out their policy on accepting individuals with felonies on their record.