Are planes infinite..

Like with vectors and planes.

Say, if the equation of a plane is = [3, 0, 1] + t[-3, 1, -2] + s[1, -1, -1].

Does this just tell you the orientation of the plane or do the planes only extend so far?

The s and t are so called 'free parameters',
plug in any value and you will get a point somewhere on the line.

since s,t are infinite real then YES the planes must have infinite extent in 3D space.

Direction (orientation) of plane is defined by its normal vector N:

we need three points on plane to get two intersecting lines, the vector directions then X-multiply to give N i.e. Vector Cross Product say N =AxB

P = [3, 0, 1] + t[-3, 1, -2] + s[1, -1, -1]. all points OP on plane:

t=0, s=0 => (3,0,1) is pnt on plane
t= 1, s=0 => (3 -3, 0 +1, 1 -2) = (0, 1, -1) is also pnt on plane
let line A = (0, 1, -1) - (3, 0, 1) = [-3, 1, -2] gives one of our required vectors, and

t= 0, s=1 similarly leads to line B = [ 1, -1, -1]

hence direction of plane N = [-3, 1, -2] x [ 1, -1, -1] = -6i -14j +2k or [-4, -14, 2]

Yes, it's just the orientation, the plane is infinite; a flat, two-dimensional mesh with infinite area.

the plane is obviously infinite. u cant limit the dimension of a plane!