I will consider another counterexample, based on Anton Anton's ideas. I think it looks simpler.

We will construct a closed domain $D$ in a plane and three functions on it with same restrictions to the boundary of the domain such that there is no choice of points on graphs of functions with pairwise parallel tangent planes.

First observation is the following: Let function $f$ on the coordinate plane depends on the coordinate $x$ only and function $g$ depends on the coordinate $y$ only. Then $df(a)=dg(b)$ if and only if $df(a)=dg(b)=0$. (It seems to me that the crucial Anton's idea is to consider functions with very degenerate gradient map, having value on a curve).

Now we take the functions $f(x,y)=-x^2$ and $g(x,y)=P_4(y)$ where $P_4$ is a polynomial of degree $4$ with two zeroes, three pairwise different critical values and leading term $y^4$.

We set the domain $D$ to be a set of solutions to an inequality $f \ge h$. It is a closed subset in a plane diffeomorphic to a unit disk. We define functions now: the first function is $f$, the second function is $g$. Third function is a function without critical points on $D$ coinciding with a restriction of $f$ (or $g$) to the boundary. Such a function exists! (it is an exercise from Morse theory. I mention here that we need a polynomial of degree 4 (degree 2 is unsufficient) to satisfy that extension without critical points property).

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I will consider another counterexample, based on Anton ideas. I think it looks simpler.

We will construct a closed domain $D$ in a plane and three functions on it with same restrictions to the boundary of the domain such that there is no choice of points on graphs of functions with pairwise parallel tangent planes.

First observation is the following: Let function $f$ on the coordinate plane depends on the coordinate $x$ only and function $g$ depends on the coordinate $y$ only. Then $df(a)=dg(b)$ if and only if $df(a)=dg(b)=0$. (It seems to me that the crucial Anton's idea is to consider functions with very degenerate gradient map, having value on a curve).

Now we take the functions $f(x,y)=-x^2$ and $g(x,y)=P_4(y)$ where $P_4$ is a polynomial of degree $4$ with two zeroes, three pairwise different critical values and leading term $y^4$.

We set the domain $D$ to be a set of solutions to an inequality $f \ge h$. It is a closed subset in a plane diffeomorphic to a unit disk. We define functions now: the first function is $f$, the second function is $g$. Third function is a function without critical points on $D$ coinciding with a restriction of $f$ (or $g$) to the boundary. Such a function exists! (it is an exercise from Morse theory)