On regular maps and parallel lines

Let $f:R^{m+1}\to R^{m+2^r}$, where $2^{r-1}\le m+1<2^r$, 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 $m+1≠2^{r-1}$, and satisfy $x_0≠x_1$, $y_0≠y_1$, $\{x_0,x_1\}≠\{y_0,y_1\}$ 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.