You can also use MeshFunctions
but you need to "cheat" to include the t=0 point, e.g.
This is an edited post based on the correct comments by Oska and gpap. In my original answer I used the equivalent of Function[{x,y,z,t},z] though my intention was Function[{x,y,z,t},t]. I have corrected and now labelled the points.
r[t_] := {Cos[7 Pi*t], Cos[6.2*Pi*t], 2.5*t};
Show[ParametricPlot3D[r[t], {t, -0.01, 2}, MeshFunctions -> (#4 &),
Mesh -> {{0., 1}}, MeshStyle -> {PointSize[0.03], Red}],
Graphics3D[
MapThread[
Text[Framed[#1], r[#2], {1, -1}] &, {{"Position 1",
"Position 2"}, {0, 1}}]]]
