I believe you'll need to find the minimums and maximums for each dimension and then calculate the range for each dimension to get the size of the box that would fit the object. (Unless you're asking for how to calculate the volume?) Also, the units are whatever units you want. If you want the height of the object to be a certain size, then you'll need to multiply each dimension by the ratio of the desired height to the current height. Below is a brute-force approach to getting the dimensions. Note that for one of the dimensions I needed to use FindMinimum rather than Minimize to get the appropriate minimum. You'll need to check to see if all of the other Minimums and Maximums converged properly.

a = 1.06;
b = 6;
c = 1.5;
d = 1;
e = 1.2;
tr = 0;
tmax = 50;

(* Find maximum and mininum for each dimension *)
(* 1st dimension *)
max1 = Maximize[{a^
     t (Cos[t] (1 + 
        c (Cos[e] Cos[\[Theta]] + d Sin[e] Sin[\[Theta]]))), -50 <= 
     t <= 0 && -\[Pi] <= \[Theta] <= \[Pi]}, {t, \[Theta]}]
min1 = FindMinimum[
  a^t (Cos[t] (1 + 
       c (Cos[e] Cos[\[Theta]] + 
          d Sin[e] Sin[\[Theta]]))), {t, -0.5}, {\[Theta], 1.5}]
(* 2nd dimension *)
max2 = Maximize[{a^
     t (Sin[t] (1 + 
        c (Cos[e] Cos[\[Theta]] + d Sin[e] Sin[\[Theta]]))), -50 <= 
     t <= 0 && -\[Pi] <= \[Theta] <= \[Pi]}, {t, \[Theta]}]
min2 = Minimize[{a^
     t (Sin[t] (1 + 
        c (Cos[e] Cos[\[Theta]] + d Sin[e] Sin[\[Theta]]))), -50 <= 
     t <= 0 && -\[Pi] <= \[Theta] <= \[Pi]}, {t, \[Theta]}]
(* 3rd dimension *)
max3 = Maximize[{a^
     t (b + c (Cos[\[Theta]] Sin[e] - d Cos[e] Sin[\[Theta]])), -50 <=
      t <= 0 && -\[Pi] <= \[Theta] <= \[Pi]}, {t, \[Theta]}]
min3 = Minimize[{a^
     t (b + c (Cos[\[Theta]] Sin[e] - d Cos[e] Sin[\[Theta]])), -50 <=
      t <= 0 && -\[Pi] <= \[Theta] <= \[Pi]}, {t, \[Theta]}]

(* Construct object *)
g = ParametricPlot3D[
   a^t {Cos[
       t] (1 + c (Cos[e] Cos[\[Theta]] + d Sin[e] Sin[\[Theta]])), 
     Sin[t] (1 + c (Cos[e] Cos[\[Theta]] + d Sin[e] Sin[\[Theta]])), 
     b + c (Cos[\[Theta]] Sin[e] - 
         d Cos[e] Sin[\[Theta]])}, {\[Theta], -Pi, Pi}, {t, -tmax, 0},
    PlotStyle -> 
    Directive[Opacity[0.5], Lighter[Yellow, .3], 
     Specularity[White, 10], Thickness[.1]], Axes -> True, 
   ViewPoint -> {2, 2, -6}, Boxed -> True, 
   PlotRange -> {{-3, 3}, {-3, 3}, {0, 7.5}}, ImageSize -> {400, 450},
    Mesh -> None, PlotPoints -> 400, 
   MaxRecursion -> ControlActive[0, Automatic], 
   RotationAction -> "Clip", SphericalRegion -> True];

(* Construct points where minimums and maximums occur *)
{max11, max12, 
  max13} = {max1[[1]], 
   a^t (Sin[
       t] (1 + c (Cos[e] Cos[\[Theta]] + d Sin[e] Sin[\[Theta]]))), 
   a^t (b + c (Cos[\[Theta]] Sin[e] - d Cos[e] Sin[\[Theta]]))} /. 
  max1[[2]]
{min11, min12, 
  min13} = {min1[[1]], 
   a^t (Sin[
       t] (1 + c (Cos[e] Cos[\[Theta]] + d Sin[e] Sin[\[Theta]]))), 
   a^t (b + c (Cos[\[Theta]] Sin[e] - d Cos[e] Sin[\[Theta]]))} /. 
  min1[[2]]
{max21, max22, 
  max23} = {a^
     t (Cos[t] (1 + 
        c (Cos[e] Cos[\[Theta]] + d Sin[e] Sin[\[Theta]]))), 
   max2[[1]], 
   a^t (b + c (Cos[\[Theta]] Sin[e] - d Cos[e] Sin[\[Theta]]))} /. 
  max2[[2]]
{min21, min22, 
  min23} = {a^
     t (Cos[t] (1 + 
        c (Cos[e] Cos[\[Theta]] + d Sin[e] Sin[\[Theta]]))), 
   min2[[1]], 
   a^t (b + c (Cos[\[Theta]] Sin[e] - d Cos[e] Sin[\[Theta]]))} /. 
  min2[[2]]
{max31, max32, 
  max33} = {a^
     t (Cos[t] (1 + 
        c (Cos[e] Cos[\[Theta]] + d Sin[e] Sin[\[Theta]]))), 
   a^t (Sin[
       t] (1 + c (Cos[e] Cos[\[Theta]] + d Sin[e] Sin[\[Theta]]))), 
   max3[[1]]} /. max3[[2]]
{min31, min32, 
  min33} = {a^
     t (Cos[t] (1 + 
        c (Cos[e] Cos[\[Theta]] + d Sin[e] Sin[\[Theta]]))), 
   a^t (Sin[
       t] (1 + c (Cos[e] Cos[\[Theta]] + d Sin[e] Sin[\[Theta]]))), 
   min3[[1]]} /. min3[[2]]

(* Graph object and maximum and minimum points *)
p = Graphics3D[{PointSize[0.05], 
    Point[{{max11, max12, max13}, {min11, min12, min13}, {max21, 
       max22, max23}, {min21, min22, min23}, {max31, max32, 
       max33}, {min31, min32, min33}}]}];
Show[{g, p}]

(* Object dimensions *)
{max1[[1]] - min1[[1]], max2[[1]] - min2[[1]], max3[[1]] - min3[[1]]}