In order for two lines to be parallel, they should have the same slope and different y-intercepts. Since you want a line parallel to y=2x+2, you know you're looking for a line whose slope is 2; all we have to do is determine what y-intercept, b, will give us something that passes through the point (-1,3).
So take y=2x + b, plug in the point (-1,3) to get 3 = 2(-1) + b, and in order for this to be true, b=5.
So your line should be y = 2x + 5, which is parallel to the original line and passes through the desired point. :)