Odds to clone a parent
Assumptions

You are ignoring recombination and mutations. If you were to add these things, then this probability would be extremely smaller. In apparence, it would be infinitely smaller. It makes no biological sense to make such assumption but for the sake of the argument, let's assume only segregation happen. We will also assume that the parents are completely unrelated to start with (see @BryanKrause 's comment below).

For on given pair of chromosome

First, for a given chromosome number, both parents must each transmit a different chromosome to each offspring. This happens with probability $\frac{1}{2}$ per parent and therefore $\frac{1}{4}$ for both parents. Then, if the brother transmit the maternally derived chromosome, then the sister must transmit the maternally derived chromosome as well (and vice-versa). This occurs with probability $\frac{1}{2}$. So for a given pair of chromosome the probability to clone the parent pair of chromosomes (assuming no recombination and no mutation) is $\frac{1}{4}\frac{1}{2} = \frac{1}{8}$.

For the whole genome

The probability of this to happen for all 23 pairs of chromosome is $\frac{1}{8}^{23} = 1.69 \cdot 10^{-21}$, that is one chance out of a 1.69 thousands billions billions.

As you ask for the odds, note that for such low probability the odds ratio is pretty much equal to the probability.