To identify the incorrect statement, let's analyze each option :

Option A : "In manganate and permanganate ions, the π-bonding takes place by overlap of p-orbitals of oxygen and d-orbitals of manganese." This statement is true. In both manganate ${MnO}_4^{2-} $ and permanganate ${MnO}_4^- $ ions, the manganese atom is in a high oxidation state (+6 and +7, respectively) and forms bonds with oxygen atoms involving the overlap of p-orbitals of oxygen and d-orbitals of manganese.

Option B : "Manganate ion is green in colour and permanganate ion in purple in colour." This statement is true. Manganate ions are indeed green and permanganate ions are purple in aqueous solution.

Option C : "Manganate and permanganate ions are paramagnetic." This statement is incorrect. Both manganate and permanganate ions are diamagnetic, not paramagnetic. In these ions, manganese is in +6 and +7 oxidation states, respectively, and has no unpaired electrons.

Manganate($$\mathop {Mn}\limits^{ + 6} $$O₄ ^2–) ,

Mn(25) = [Ar]3d⁶4s²

Mn⁺⁶(25) = [Ar]3d¹ $$ \to $$ Paramagnetic

Permanganate ($$\mathop {Mn}\limits^{ + 7} $$O₄^-)

Mn(25) = [Ar]3d⁶4s²

Mn⁺⁷(25) = [Ar]3d⁰ $$ \to $$ Diamagnetic

Option D : "Manganate and permanganate ions are tetrahedral." This statement is true. Both manganate and permanganate ions have a tetrahedral geometry, with the manganese atom at the center and the four oxygen atoms surrounding it.

Therefore, the incorrect statement is option C.