Axxis moving parallel to another

I'm not sure I am understanding you correctly, but no, I don't think what you're describing is like Lorentz transformation at all.

First, just to be clear, you are talking about an alternate coordinate system in the same 2-D space, correct? Both it and the original coordinate system share the same x-axis, but have different parallel y axes - again, correct? If this is the case, then what you've described is a second coordinate system that is related to the first by a translation, specifically a translation in the x-direction.

Lorentz transformations are more akin to rotations, and thus are not really similar to what you've described. They are not ordinary rotations, however, in that they have imaginary rotation angles (that's one way to describe them, anyway), and the axes move opposite directions. For example, if the Lorentz transformation to the rest frame of an object moving in the x-direction corresponded to drawing the x-axis rotated in a counter-clockwise direction, then the t-axis would rotate in the clockwise direction.