What is the equation of a straight line parallel to the y-axis?

$m=\frac10$ is a nonsensical expression. As such, you should come as no surprise that anything you write after that expression will also not make any sense.

When the line is vertical, it's quite inaccurate to speak of it's slope as being equal to $\frac10$. It helps to be accurate when talking about mathematical terms, and to keep in mind what the definitions of what you are talking about are.

The definition of the slope of a line is that it is the value $k$ for which the equation $$y=kx + n$$ describes the line. In more detail, it's the value for which the set $$\{(x, y)| y=kx + n\}$$ is equal to the line.

If the line is vertical, it is relatively easy to show that no such value $k$ exists, and therefore, by the definition of the slope, the line does not have a slope. The slope is not equal to $\frac10$.