We say that a random variable $X$ is subexponential with constant $c$ if $$ \mathbb{E}[\exp(t(X-p))] \le \exp(c^2t^2), \qquad |t|\le 1/c. $$
If instead the MGF were upper-bounded by $\exp(c^2t^2)$ for all $t\in\mathbb{R}$, we would be talking about the subgaussian constant, and for the Bernouilli $X\sim Ber(p)$; it is known that the optimal constant is $ c^2=\frac{1-2p}{4\log((1-p)/p)} $.
The optimal subexponential constant must be known as well; what is it?
