  We study moduli spaces of mirror non-compact Calabi-Yau threefolds enhanced
with choices of differential forms. The differential forms are elements of the
middle dimensional cohomology whose variation is described by a variation of
mixed Hodge structures which is equipped with a flat Gauss-Manin connection. We
construct graded differential rings of special functions on these moduli spaces
and show that they contain rings of quasi-modular forms. We show that the
algebra of derivations of quasi-modular forms can be obtained from the
Gauss--Manin connection contracted with vector fields on the enhanced moduli
spaces. We provide examples for this construction given by the mirrors of the
canonical bundles of $\mathbb{P}^2$ and $\mathbb{F}_2$.
