In general, Reduce
is the most powerful Mathematica function for solving equations.
Reduce[par == line && 0 < u < 2 Pi && -Pi < v < Pi, {t, u, v}, Reals]
(* (t == AlgebraicNumber[Root[-18291750000 + 235480000*#1 - 237800*#1^2 +
48*#1^3 + #1^4 & , 1, 0], {0, 1/725, 0, 0}] &&
u == 2*ArcTan[AlgebraicNumber[Root[-18291750000 + 235480000*#1 -
237800*#1^2 + 48*#1^3 + #1^4 & , 1, 0],
{3961/4536, -2479/1315440, 1121/353220000, 83/19073880000}]] &&
v == 2*ArcTan[AlgebraicNumber[Root[-18291750000 + 235480000*#1 -
237800*#1^2 + 48*#1^3 + #1^4 & , 1, 0], {3505/1484, -9221/6455400,
17053/9360330000, 197/18720660000}]]) ||
(t == AlgebraicNumber[Root[-18291750000 + 235480000*#1 - 237800*#1^2 +
48*#1^3 + #1^4 & , 2, 0], {0, 1/725, 0, 0}] &&
u == 2*ArcTan[AlgebraicNumber[Root[-18291750000 + 235480000*#1 -
237800*#1^2 + 48*#1^3 + #1^4 & , 2, 0],
{3961/4536, -2479/1315440, 1121/353220000, 83/19073880000}]] &&
v == 2*ArcTan[AlgebraicNumber[Root[-18291750000 + 235480000*#1 -
237800*#1^2 + 48*#1^3 + #1^4 & , 2, 0],
{3505/1484, -9221/6455400, 17053/9360330000, 197/18720660000}]]) @)
% // N
(* (t == -1.07503 && u == 2.29164 && v == -0.761533) ||
(t == 0.116633 && u == 1.27311 && v == 2.30858) *)
The intersections can be visualized by
Show[
ParametricPlot3D[par, {u, 0, 2 Pi}, {v, -Pi, Pi},
PlotStyle -> Opacity[.5], LabelStyle -> {15, Bold, Black}],
ParametricPlot3D[line, {t, -2, 1}, PlotStyle -> {Black, Thick}],
ListPointPlot3D[{line /. t -> -1.07503, line /. t -> 0.11663}, PlotStyle -> Red]]

Addendum: Use FindRoot
In the event that Reduce
does not provide a solution, FindRoot
almost always will, but requires multiple initial guesses to obtain multiple intersections, as is the case here.
FindRoot[par == line, {t, 0}, {u, Pi}, {v, 0}]
FindRoot[par == line, {t, 0}, {u, Pi}, {v, 2}]
(* {t -> -1.07503, u -> 2.29164, v -> -0.761533} *)
(* {t -> 0.116633, u -> 1.27311, v -> 2.30858} *)
par = 5 {Cos[u], Cos[v] + Sin[u], Sin[v]}; line = {1, 2, 3} + {4, -5, 6} t; Solve[par == line && 0 < u < 2 \[Pi] && -\[Pi] < v < \[Pi], {t, u, v}, Reals, Method -> Reduce]
$\endgroup$