Hint: consider what happens when you make more assumptions about the specific distribution of the errors. Then you can write down exact conditional densities. After multiplying a few together, you will have the joint density of all the time observations, and strong stationarity deals with this joint distribution.
For your model: $$ p(y_1, y_2, \ldots , y_n) = \prod_{t=3}^n p(y_t \mid y_{t-1}, y_{t-2} ) p(y_1, y_2). $$ If you assumed that the errors were Normally distributed then $$ p(y_t \mid y_{t-1}, y_{t-2} ) = N. $$