But the genes don't get all mixed together like that. You have 23 pairs of chromosomes, each with a bunch of genes in them. In each pair, you inherited one chromosome from your mother, and one from your father. There might be some genetic drift between one and the other, but it would be the exception, not the rule.
However, that's not the end of the story. When you get conceived, as a fertilized egg, you indeed have 23 pairs of chromosomes, one from each parent, so, say, a chromosome 2F from your father and a chromosome 2M from your mother (if you have more than two chromosomes #2,
one of your parents probably was a chimpanzee). During mitosis, normal cell division, in principle this is faithfully copied.
However, the production of gamets (eggs and sperm) involves another kind of cell division,
meiosis, where one cell copies its chromosomes and divides into four. During that,
genetic recombination may take place: two sister chromosomes get entangled and exchange a piece of their genetic material. So each of your gametes can have a different makeup of chromosome #2: either truthfully a copy of 2F or of 2M, or a strand of 2F followed by a bit of 2M followed again by the rest of 2F (as in the picture on the wiki page on genetic recombination), or most of 2M with a top of 2F (as in the picture on the wiki page on meiosis).
So you don't necessarily pass a whole chromosome to your children that's identical to one of those you got from a parent.