Identify the species and oxidation states

• In manganate ion, $\mathrm{MnO_4^{2-}}$: $x + 4(-2) = -2 \Rightarrow x = +6$ So Mn is in oxidation state $+6$.

• In permanganate ion, $\mathrm{MnO_4^-}$: $x + 4(-2) = -1 \Rightarrow x = +7$ So Mn is in oxidation state $+7$.

Check magnetic nature

Mn has atomic number $25$: $\mathrm{Mn} = [Ar]3d^5 4s^2$

• For $\mathrm{Mn^{6+}}$: remove $2$ electrons from $4s$ and $4$ from $3d$: $\mathrm{Mn^{6+}} = [Ar]3d^1$ This has one unpaired electron, so $\mathrm{MnO_4^{2-}}$ is paramagnetic.

• For $\mathrm{Mn^{7+}}$: remove $2$ electrons from $4s$ and $5$ from $3d$: $\mathrm{Mn^{7+}} = [Ar]3d^0$ This has no unpaired electrons, so $\mathrm{MnO_4^-}$ is diamagnetic.

Therefore, the statement "Manganate and permanganate ions are paramagnetic" is incorrect, because only manganate is paramagnetic, while permanganate is diamagnetic.

Check the other statements

Option A: In standard JEE-level inorganic chemistry, the $\pi$-bonding in oxoanions such as manganate and permanganate is described as overlap of filled $p$-orbitals of oxygen with vacant $d$-orbitals of manganese ($p\pi-d\pi$ bonding). So this statement is taken as correct.

Option B:

• $\mathrm{MnO_4^{2-}}$ is green
• $\mathrm{MnO_4^-}$ is purple/violet

Hence, correct.

Option D: Both $\mathrm{MnO_4^{2-}}$ and $\mathrm{MnO_4^-}$ have four bonding pairs around Mn and no lone pair on the central atom, so both are tetrahedral. Hence, correct.

Conclusion

The incorrect statement is: $\boxed{\text{C}}$