The focus of this article is on the different behavior of large deviations of random functionals associated with the parabolic Anderson model above the mean versus large deviations below the mean. The functionals we treat are the solution u(x, t) to the spatially discrete parabolic Anderson model and a functional A (n) which is used in analyzing the a.s. Lyapunov exponent for u(x, t). Both satisfy a "law of large numbers", with lim(t ->infinity) 1/t log u (x, t) = lambda (kappa) and lim(n ->infinity) An/n = alpha. We then think of alpha n and lambda(kappa)t as being the mean of the respective quantities A (n) and log u(t, x). Typically, the large deviations for such functionals exhibits a strong asymmetry; large deviations above the mean take on a different order of magnitude from large deviations below the mean. We develop robust techniques to quantify and explain the differences.