No need for a point. Use parameteric representation

ClearAll[x, y, z, t];

p1 = x + 5 y + 2 z + 3 == 0; (*equation of first plane*)

p2 = 2 x + 3 y + 6 z + 4 == 0; (*equation of second plane*)

eq1 = Simplify[Inner[Subtract, Thread[2 *p1, Equal], p2, Equal]]; (*eliminate x*)

zt = t; (*let z=t*)

yt = y /. First@Solve[eq1 /. z -> zt, y]; (*find y in terms of t*)

xt = x /. First@Solve[p1 /. {z -> t, y -> yt}, x]; (*find x in terms of t*)

Show[

Plot3D[z /. First@Solve[p1, z], {x, -2, 2}, {y, -3, 3},PlotStyle -> Green, Mesh -> None],

Plot3D[z /. First@Solve[p2, z], {x, -2, 2}, {y, -3, 3},PlotStyle -> LightBlue, Mesh -> None],

ParametricPlot3D[{xt, yt, zt}, {t, -3, 3}, PlotStyle -> {Thickness[.03], Red}], Axes -> None, Boxed -> False

]