jmc

Setting $x=0$ in both equations, we get $$2013=3h^2+j\text{and}2014=2h^2+k.$$Solving for $j$ and $k,$ we can rewrite the given equations as $$y=3(x-h{)}^2+(2013-3h^2)\text{and}y=2(x-h{)}^2+(2014-2h^2),$$or $$y=3x^2-6xh+2013=3(x^2-2hx+671)\text{ and }y=2x^2-4hx+2014=2(x^2-2hx+1007).$$The left equation has positive integer roots, which must multiply to $671$ and sum to $2h.$ Similarly, the right equation has positive integer roots, which must multiply to $1007$ and sum to $2h.$ Since $671=61\cdot 11$ and $1007=19\cdot 53,$ we see that $$2h=61+11=19+53=72,$$so $h=\boxed{36}.$