This may be overkill, but I think it is interesting to use RegionIntersection
here.
plane1 =
InfinitePlane[{x, y, z} /.
FindInstance[
2 x + y + z == 1 && {x, y} ∈ Rectangle[], {x, y, z},
Reals, 3]]
plane2 = InfinitePlane[{x, y, z} /.
FindInstance[
3 x - 2 y - z == 5 && {x, y} ∈ Rectangle[], {x, y, z},
Reals, 3]]
line = RegionIntersection[
plane1, plane2
]
Graphics3D[{Red, plane1, plane2, Blue, line}]
(* InfinitePlane[{{31/302, 1, -(31/151)}, {1, 1, -2}, {33/302, 0, 118/151}}] *)
(* InfinitePlane[{{31/302, 1, -(2021/302)}, {1, 1, -4}, {33/302, 0, -(1411/302)}}] *)
(* InfiniteLine[{7/5, 1, -(14/5)}, {-(1/5), -1, 7/5}] *)

Now you have your line as an InfiniteLine
object, you can convert this to a parametric form via the formulas here.
{point, vector} = List @@ line;
equation = point + vector t
(* {7/5 - t/5, 1 - t, -(14/5) + (7 t)/5} *)
Which I can show graphically is equivalent to the form OP seeks,
ParametricPlot3D[{
equation,
{1 + t, 5 t - 1, -7 t}
}, {t, 0, 5}]
