Why is portfolio optimization a convex problem if variance is concave?

First, a correction is in order: the math question you cite is the variance for a Bernoulli random variable as a function of the parameter $p$. That is, indeed, concave in $p$. However, the variance of a portfolio, $w^TSigma w$, is not concave in $w$. So your initial presumption of concavity is not correct.

For a Bernoulli random variable, the uncertainty of outcomes is most uncertain for outcomes that are equally likely. That is very different from a portfolio where weights of $1/N$ diversify our exposure to multiple sources of risk and thus tend to reduce the total variance.

For a mean-variance portfolio optimization, we have the following problem: $$ begin{align} max_w &~w^T R - frac{lambda}{2} w^TSigma w \ text{s.t.} &~||w||_1 = 1. end{align} $$

Here, the objective function is a linear function minus a quadratic form; that is concave.

If we instead use a coherent measure of risk, the objective function just becomes $w^T R - frac{lambda}{2} text{Risk}(w)$. Note that coherent risk measures (like CVaR/ES/TCE/ETL) are convex as discussed in Föllmer and Schied (2008).

Both of these objectives are concave. However, as Arshdeep's existing answer notes, a concave function can be made convex by multiplying by -1. Finally I should note that we do not even need convexity but often merely quasi-convexity (which might be the case for constrained optimizations).