Indeed a rectangle of width 144 would have every horizontal line from 0 to 144; neither 0 nor 144 would count since both are on the boundary, so there would be 143 lattice points per line.

Similarly if the angle of the parallelogram were, say, 45 degrees, the oblique lines would be running through each lattice point along the way, so again there would be 143.

But when the oblique lines, as they pass y=1, y=2, and so on - when they pass between lattice points, then one more lattice point will get counted in each line, so 144 is the answer.

I've made an illustrative diagram on Desmos with 3 parallelograms and a rectangle, but with width 2 instead of 144.  The number of interior lattice points per line varies depending on the relative prime-ness of delta y vs delta x.

https://www.desmos.com/calculator/4gsqgutjis