Is it possible to deduce facts about a person's parents just by studying his/her genome?
Not the kind of complex phenotype that you describe (because nobody knows for example if/how "being abusive" is written in the genome), but yes, some things can be determined.

The easiest is through sex chromosomes. 

For example: if you look at the X chromosome of a man, you can tell for sure that it comes from his mother (because the Y can only come from the father, quite logically) so any allele that you see there is also part of the mother's genome - but since she has a second copy, maybe it does not translate into a phenotype. 

Sometimes you can even guess the phenotype of one of the parents: if you study a woman's X chromosomes and see that both have alleles that cause color-blindness (encoded on this chromosome) it means that the only X chromosome of the father has this allele, so the father is probably color-blind. 

See: https://ghr.nlm.nih.gov/condition/color-vision-deficiency