Answer : $\large\frac{2}{3}$
It is given that the parabola passes through the point $(3,2)$
Length of its latus rectum is $2a$
$y^2=4ax$
(ie) $(2)^2=4a(3)$
$\Rightarrow a=\large\frac{1}{3}$
$\therefore$ Length of the latus rectum is $2\times \large\frac{1}{3}=\frac{2}{3}$