The equation $a{x}^2+2hxy+b{y}^2+c=0$ , $ab\ne 0$ represents a pair of real and parallel lines if and only if 

$a{x}^2+2hxy+b{y}^2+c=0$ represents a pair of parallel lines (distinct or coincident)

If $ab-h^2=0$

Clearly, $ab>0$, if $a>0$ then the equation can be written as ${\left(\sqrt{a}x\pm \sqrt{b}y\right)}^2+c=0\Rightarrow c\le 0$

For real solution.

If $a<0$ then the equation can be written as ${\left(\sqrt{-a}x\pm \sqrt{-b}y\right)}^2-c=0\Rightarrow c\ge 0$ for the real solution thus a and c must be of opposite signs. Also a and b are of the same sign so b and c must be of opposite signs.

Hence $ac\le 0$ or $bc\le 0$.