Answer
Neither parallel nor perpendicular.
Work Step by Step
We know that two parallel lines have the same slope ($m_1=m_2$).
We also know that perpendicular lines have negative reciprocal slopes ($m_1=-\frac{1}{m_2}$).
A line in slope-intercept form has the equation:
$y=mx+b$ ($m=slope$, $b=y-intercept$)
We put the line equations into slope-intercept form:
$2x+y=6$
$y=6-2x$
$y=-2x+6$
$x-y=4$
$y=x-4$
$y=1x-4$
The two slopes ($-2$ and $1$) are not the same and not negative reciprocals. Thus the lines are neither parallel nor perpendicular.