$\because BD$是$\odot O$的直径，$\angle CBD=30^{\circ}$，

$\therefore \angle BCD=90^{\circ}$，

$\therefore \angle D=90^{\circ}-30^{\circ}=60^{\circ}$，

$\therefore \angle A=\angle D=60^{\circ}$.

故选$C$.