Two famous manifestations of local rigidity for higher rank abelian actions are 1) KAM-rigidity of simultaneously Diophantine torus translations (Moser) and 2) smooth rigidity of hyperbolic or partially hyperbolic higher rank actions (Damjanovic and Katok). To complete the picture of local rigidity for higher rank abelian affine actions on the torus, the case of parabolic actions must be addressed.   With D. Damjanovic and M. Saprykina we show that KAM-rigidity of abelian parabolic actions holds when one of the elements of the action is of step 2. With S. Durham, we show that KAM-rigidity does not necessarily hold if the elements of the action are all of step higher than 2.    \begin{definition} We say that a linear map $A\in \SL(d,\Z)$ is parabolic of step $n$ if  $ (A-\Id)^n=0, $ and $ (A-\Id)^{n-1}=0. $ An affine map $a(\cdot )=A(\cdot) +\al$ is said to be of step $n$ if $A$ is of step n.  We say that a $\Z^2$ affine action by parabolic elements is of step $n$  if all of its elements are of step at most $n$.  \end{definition}