  We explicitly prove the quantum modularity of partial theta series with even
or odd periodic coefficients. As an application, we show that the
Kontsevich-Zagier series $\mathscr{F}_t(q)$ which matches (at a root of unity)
the colored Jones polynomial for the family of torus knots $T(3,2^t)$, $t \geq
2$, is a weight $3/2$ quantum modular form. This generalizes Zagier's result on
the quantum modularity for the "strange" series $F(q)$.
