P, Q lie on the x-axis and R, S lie on the y-axis. Maximum number of parabolas that can be drawn through these points is 

 Let the equation of PR and QS be  $ax+by+c=0$  and  $a^{1}x+b^{1}y+c^1=0$ respectively.
Now, any conic through points P, Q, and R & S is  $(ax+by+c)(a^{1}x+b^{1}y+c^1)+\lambda xy=0$
$\Rightarrow (a{a}^1{x}^2+b{b}^1{y}^2+a{b}^{1}xy+b{a}^{1}xy-\lambda xy)+\mathrm{........}=0$
This represents parabola .Then 2^nd degree terms form perfect square.
So maximum two values of  $\lambda$  exist