The aim of the present paper is to investigate the Hyers-Ulam stability of the Pexiderized quadratic functional equation, namely of $f(x+y)+f(x-y)=2g(x)+2h(y)$ in paranormed spaces. More precisely, first we examine the stability for odd and even functions and then we apply our results to prove the Hyers-Ulam stability of the quadratic functional equation $f(x+y)+f(x-y)=2f(x)+2f(y)$ in paranormed spaces for a general function.