I think you posted this before, and probably got no response.
You asked for "the" line parallel to the plane
2x+6y+4z = 1.
There is no unique line.
Think of the floor of your kitchen as 2x+6y+4z = 1 and you have a table whose top is parallel to the floor and contains the point (3,2,1).
Laying down a ruler would be a line parallel to the plane and passing through the point.
How many such positions could you choose ?
a normal to your given plane is (2,6,4) or (1,3,2)
so any vector whose dot product with the above normal would be parallel to the plane
e.g. (-5,1,1)∙(1,3,2) = -5+2+3 = 0
and (1,-1,1)∙(1,2,3) = 1-3+2 = 0 are just two such vectors
let's take (-5,1,1)
so the equation of one such line is
x = 3 -5t
y = 2 + t
z = 1 + t
thank you :)