The correct answer is $\boxed{0}$.
To find $\frac{{{\partial ^2}{\text{f}}}}{{\partial {\text{x}}\partial {\text{y}}}}$, we can use the following formula:
$$\frac{{{\partial ^2}{\text{f}}}}{{\partial {\text{x}}\partial {\text{y}}}} = \frac{\partial}{\partial y} \left[ \frac{\partial f}{\partial x} \right]$$
In this case, we have $f(x, y) = yx$. Therefore,
$$\frac{\partial f}{\partial x} = y$$
and
$$\frac{\partial^2 f}{\partial x \partial y} = \frac{\partial}{\partial y} \left[ y \right] = 0$$
Plugging in $x = 2$ and $y = 1$, we get
$$\frac{{{\partial ^2}{\text{f}}}}{{\partial {\text{x}}\partial {\text{y}}}} \bigg|_{x = 2, y = 1} = 0$$
Therefore, the correct answer is $\boxed{0}$.
The other options are incorrect because they do not give the correct value of $\frac{{{\partial ^2}{\text{f}}}}{{\partial {\text{x}}\partial {\text{y}}}}$ at $x = 2$ and $y = 1$.