Parseval Equality says, Fourier transformation, as a linear map from one function space (function on x), to another function space (function on p), preserves 'norm'. Norm is just a fancy way of saying 'length of a vector'.

What do we mean by the length of a function?

Continuous Fourier transformation (OK, I switched to Boas convention) $f(x) = \int_\R F(p) e^{ipx} dp.$ $F(p) = (1/2\pi) \int_\R f(x) e^{-ipx} dx.$