Answer: yes, one can.
Suppose that if we replace each 0 in the decimal fraction $d$ by 1 we obtain the decimal fraction $a$, and if we replace each 1 in the decimal fraction $d$ by 2 we obtain the decimal fraction $b$. Since both the fractions $a$ and $b$ are periodic, we see that both these numbers are rational numbers.
If we replace each 2 in the decimal fraction $a$ by 0 we obtain some decimal fraction $\alpha$ which is evident periodic and so $\alpha$ is a rational number. Similarly, if we replace each 0 in the decimal fraction $b$ by 1 we obtain some decimal fraction $\beta$ which is also periodic, and so $\beta$ is a rational number too.
It is easy to see that $d = \beta - \alpha$. Therefore, $d$ is a rational number and so the decimal fraction $d$ is periodic.