jmc

Solving for $x$ in $3x+5y+k=0,$ we get $$x=-\frac{5y+k}{3}.$$Substituting into $y^2=24x,$ we get $$y^2=-40y-8k,$$or $y^2+40y+8k=0.$ Since we have a tangent, this quadratic will have a double root, meaning that its discriminant will be 0. This give us $40^2-4(8k)=0,$ so $k=\boxed{50}.$