I am not working with trios myself - so I am just thinking out loud.

Since the child may have novel variants and the parents may also share variants, could we always correctly assign a variant to either parent? Many times would possible, but other times not.

I believe that the order 0|1 vs 1|0 has meaning only in terms of phasing. Every variant in a block lists the genotypes to be in sync with other variants in that block. That is the sole purpose of the order. For unphased variants, the order is always low to high 0/1

To figure out which parent shares the variant look at the parent genotype.