This is a followup question to the answer given to a similar question where the velocity is given as
$$v(r) = \left(1 - \frac{r_s}{r}\right)\sqrt{\frac{r_s}{r}}c \tag{1}$$
where $r$ is the radius from the Schwarzschild co-ordinates and $r_s$ is the Schwarzschild radius.
What would be the velocity depending on $t$, the co-ordinate time, i.e.
$$ \frac{dr(t)}{dt} = \;?$$
Looking at the Wikipedia link given in the referenced answer, section "Local and delayed velocities" I would guess it is
\begin{align} \frac{dr(t)}{dt} &= \frac{dr}{d\tau}\frac{d\tau}{dt}\\ &= v_{\parallel}\gamma\sqrt{1-r_s/r}\cdot\frac{1}{\gamma} \sqrt{1-r_s/r}\\ &= v_{\parallel}(1-r_s/r) \end{align}
but I can't find what $v_{\parallel}$ is.