The basic idea is that internally, if an old-style nonparametric CAD program calculates some quantity (say, the length of something-or-other), it then stores the resulting quantity. A parametric CAD program stores the recipe for performing the calculation.
So instead of a variable in your design being stored as, say "the length is 100mm", it's stored as "the length is equal to the length of bolt A minus the width of nut B". This means that if you resize bolt A, any part of the design that was calculated from its dimensions changes appropriately, rather than you having to go back and change it yourself. In both programs, there is a feature to calculate the length (100mm) from the length of bolt A, and the width of nut B; the difference is that the parametric program remembers how it calculated it.
(My understanding of this comes largely from talking to someone who had worked as a programmer at Parametric Technology, which was the company that basically invented the concept.)