To answer this in a simplified form, I will assume that an undesirable genetic trait is caused by a single gene and is not sex-linked.
If the mutant gene is dominant, then the parent will already be displaying the trait. If the child also displays that trait then any offsring have a 75% chance of inheriting it. If the child does not display the trait then the chance of offspring inheriting it is 50%.
In such cases, the parent would be aware of the defect in the family.
Now to consider recessive mutants. Two copies of the mutant gene must come together before the trait is displayed so the parent would not know if they were a carrier or not. Where that gene is rare in the population, the chance of two coming together by chance would be rare. With a cross between parent and child, there is already a 50% chance of the child carrying that gene. Any offspring prom a parent F1 cross will have a chance of 1 in 8 of displaying an otherwise unseen defect not previously seen in the family.
That applies to ANY mutant carried by the parent. Considering the huge number of genes carried in the human genome, there is a high likeihood that the parent carries not just one mutant gene but several. There is therefore a much higher chance of offspring showing at least one of those traits.
The actual probability of passing on a defect also depends upon its prevalence in the population and the chance of the parent carrying it in the first instance therefore depends on that.