Just sneaking in for a tiny bit of clarification, Michael...
"When the parents donate their 22 non-sex chromosomes (called Autosomes) each pair recombine DNA in a random manner, thus while we get half of each parent's chromosomes, we only get approximately half of our DNA from each parent because of recombination."
If we remove the sex chromosomes from the equation and look only at the autosomes (and barring any genetic structural abnormality like aneuploidy) we get exactly 50% of our DNA from each parent. It can't happen any other way because the gamete each generates contains a full, haploid chromosome.
I think where some confusion might come in is that recombination, technically "crossing over," doesn't happen when the parents' haploid chromosomes pair up after fertilization. Crossing over happens before the gamete is ever produced. In fact, in women, crossing over happens in all the oocytes the mother will ever have while she, herself is still in the womb.
It takes place during Prophase I of meiosis, and is what makes daughter cells genetically unique from one another. From there, the process enters Metaphase I, then Anaphase I, and Telophase I. Crossing over can't happen again after Prophase I is complete. Meiosis repeats those stages as Prophase II, Metaphase II, and so forth before final creation of the gamete. In the oocyte, meiosis is paused after Prophase I until puberty, and again at Metaphase II until the ovum is fertilized.
So we do get exactly half our autosomal DNA from each parent, but what we receive won't be equal 25% parts from our grandparents. That's when the variability starts.
That's also why, other than identical (monozygotic) twins, no two siblings will have the same autosomal DNA. The way 23andMe reports sibling sharing, they state the possible range as being from 38% to 61%. For a little fun with math, if we say that full siblings will share on average 50% of their half-identical region (HIR) DNA, we can estimate with a simple equation, in aggregate and roughly, how much of the parents' genomes will be passed down based on the total number of children they have: 1-(0.5n) where n is the total number of children. So two children would account for about 75% of the parents' autosomal DNA; four children, 93.75%; and we'd need seven children to get over the 99% mark.