If they are parallel they will have the same slope , m
So in y = mx + c, if there are two equations which both have the same m value they will be parallel.
If the lines are perpendicular they'll have slopes like this: 1/2 to -2/1 - where they flip and a negative gets added.
In the equations: 10x + 5y = -5 , and y = -2x + 6
We can rearrange 10x + 5y = -5 to be in the form y = mx + c
10x + 5y = -5
5y = -5 - 10x
y = -1 - 2x
y = -2x - 1
Since y = -2x - 1 and y = -2x + 6 both have the same slope of -2 they are parallel!