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This is a very active area of methods development, so we can expect these references to be quickly outdated.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