Slowness: putting space and time on *really* equal footing

If space and time are completely similar and must be treated as such everywhere, why we always have position as a function of time and never the other way around? 

$$ S_{\alpha}:= \frac{dt}{dx^\alpha}$$

$$ds^2=-b^2(x)c^2 dt^2 + a^2(t)dx^2$$

$$\tau(x)=b(x)t(x)$$