How to prove using symmetry that electrostatic force acting parallel to the line joining two charges? 
The unit vector $\hat{r}_{21}$ shows that the force is parallel to the line joing the charges. It could be otherwise unless space itself has some built-in directional property. . . This can be predicted by logical arguments from symmetry. - Berkely Physics Course Vol. 2 by Edward M Purcell.

Now, the question seems to be totally nonsense; but I do actually want to know why is the force acting parallel to the line joining charges. Or what led to assume(??) that the force is acting in that direction; it could be anywhere acting. How can I use symmetry to prove this? Plz help. 
 A: Start from the assumption that space is isotropic (independent of direction). Under that assumption, if you rotate your imagined empty-universe-with-two-charges by any angle around the line joining the two charges, then you will wind up with exactly the same physical system you started with. In particular, the charge configuration is unchanged.
Now, if the electric force on one of the charges has a component perpendicular to the line, then that component will be affected by the rotation, and thus the force acting on the charge will be different after the rotation than before it. But this means that two different forces ($\vec{F}_\text{after}$ and $\vec{F}_\text{before}$) are produced by the exact same physical system. There is a uniqueness theorem in electromagnetism that says this is impossible: any given configuration of charges produces exactly one electromagnetic field. So either space is not isotropic, or the electric force on each charge has no component perpendicular to the line.
