As an example, suppose Anne had abusive parents. Is it theoretically possible to deduce this from her genome even if she didn't inherit this quality (of being an abusive parent)?
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.
It might seem pernickety but you often can't deduce from a genome; you can only infer from it. For many characteristics about a person, there are only rough, probabilistic associations between genotype and phenotype. Not one-to-one relationships.
You can take an educated guess that someone with a certain genotype could be a social person of European ethnicity with a low risk of psychosis, which might suggest things about their parents. But there are likely many genes that influence those characteristics and still more non-genetic factors. So you couldn't be certain.
For a factor like whether the persons parents had abusive personalities, I think the genetic differences would be so subtle (if existent) and there would be so many other factors (such as the habits and choices of the parents) that you would be very unlikely to be able to draw any conclusive associations. Articles and studies about linking human genetics with a person's characteristics are listed below. If any of the genes in question are linked with those characteristics then the parents of someone with the gene could possibly have those genes and characteristics too.
Personality types including belligerence, charisma, cynicism, housekeeping, lack of personality, obsessive-compulsive behaviour and gullibility.
If anyone would like to suggest additions to that list, I'll happily add them.