Can someone tell me why this is happening and how to fix it? (See the Mathematica code and output below.)
Perhaps because your first plot used {t,2,4}
You can probably change that if you include AxesOrigin->{0,0} in one or all the plots if that is what you really want.
If I understand the syntax for ParametricPlot, the first plot draws the line segment from the point (0,2) to the point (0,4). That's the y-axis between y=2 and y=4.
Thanks for the AxesOrigin hint. I'll try that.
Your hint of trying AxesOrigin worked. Thanks!!