Suppose the slope of the three parallel lines is $m$. The equation of a line with slope $m$ and $y$-intercept 2 is $y=mx+2$. To find the $x$-intercept in terms of $m$, we set $y=0$ and solve for $x$. Doing this, we obtain $mx+2=0$ or $x=-\frac{2}{m}$. Similarly, the line with slope $m$ and $y$-intercept 3 has $x$-intercept $-\frac{3}{m}$. Also, the line with slope $m$ and $y$-intercept 4 has $x$-intercept $-\frac{4}{m}$. Since the sum of the $x$-intercepts of these lines is 36, then $\left(-\frac{2}{m}\right)+\left(-\frac{3}{m}\right)+\left(-\frac{4}{m}\right)=36$. Multiplying both sides by $m$, we obtain $-2-3-4=36m$ and so $36m=-9$ or $m=-\frac{1}{4}$.