What can you learn about someone's mother and father by examining their DNA?

In principle, you can reconstruct 1/2 the genome of each parent.

Each of these halves is still a "complete" genome, in the sense that it (usually) contains 23 full chromosomes (there are 23 human chromosomes). Every person has two copies of their genome, one of which is inherited from each parent. I strongly recommend reading about meiosis to learn more about this.

Each of these single genome copies is called a haplotype, and is an outcome of recombination between their own two genome copies. So neither copy looks exactly like a parent, but is rather a mixed-up version of that parent's genome.

There are a lot of genomic methods out there that try do exactly this task (reconstructing parental haplotypes) by various means. It is somewhat complicated and expensive, but possible in principle:

• One method is trio-binning, which involves also collecting data from the parents, but focusing most effort on the child. This is not very convenient of course if you don't have samples from the parents.

• Other methods use only information from the child, but rely on various genome technologies to reconstruct haplotypes inherited from each parent. Here is an early example, and here is a very recent example. These methods however do this mostly on a chromosome-by-chromosome basis, so it is not always obvious how to group the 46 total chromosomes into 2 parental haplotypes of 23 each.

This is a very active area of methods development, so we can expect these references to be quickly outdated.

I do not think that any method will be able to use only the DNA of the child to reconstruct the full genomes of the parents. Maybe if you have a large database of other genomes you can do some imputation of the missing haplotypes, but I doubt that that would work very well.