Use illustrator its by far easier. After your edits its easier to understand what your after. Basically your asking if there is a grid snap? Sure, to turn the grid on and turn grid snapping on:
- Choose View → Show Grid
- Then enable gridsnapping View → Snap to Grid
the rest should be self explanatory on account that you know how to draw the shape. But to add a bit your shape is a parallelogram. That means that you can always clone the opposite side by alt dragging selected shape to clone path parts.
Now its not so easy to do this on a grid. As the length of edges does not necessarily coincide with the grid (especially when your at an angle). When you hold Alt and click then you get a dialog for entering values for the tool. This will help you doing the drawing WITHOUT a grid. Just type in direction and line length and then clone them around.
Image 1: Time lapse of a arbitrary example
Tip: Use the line tool its conceptual easier. If you need the shape closed afterwards just use shape builder tool to make shapes.