In parametric underwater communications, nonlinear effects occurring during the propagation of intense acoustic waves through the underwater channel are used for communication purposes. Simplified for information theoretical analysis, the inherent nonlinear channel can be modelled by a second-order nonlinearity concatenated with an additive noise process. In this paper, a new capacity upper bound is derived for the simplified parametric channel in closed form by using a duality based approach. Further, a new capacity lower bound is determined by means of numerical integrations. The proposed capacity bounds are asymptotically tight for high SNRs, i.e., the approximation error vanishes if the SNR tends to infinity. Applying the bounds for performance analysis, it will be discussed that conventional signal waveforms are not advisable for parametric underwater communications from an information theoretical perspective. Instead, signal shaping in view of the channel nonlinearity increases the mutual information significantly. The outcomes are of general relevance for AWGN channels containing squaring elements.