On regular maps and parallel lines

Let $f: R^{m+1}\to R^{m+2^r}$, where $2^{r-1}\leq m+1 <2^r$, be a continuous map. Improving a recent result of Frick and Harrison, we show that there are $4$ points $x_0,\, x_1,\, y_0,\, y_1$ in $R^m$, which are distinct if $m+1\not=2^{r-1}$, and satisfy $x_0\not=x_1$, $y_0\not=y_1$, $\{ x_0, x_1\} \not=\{ y_0,y_1\}$ if $m+1=2^{r-1}$, such that the vectors $f(x_1)-f(x_0)$ and $f(y_1)-f(y_0)$ are parallel.