How do biologists determine the parents of a child
Your description is not totally correct (or understandable). There is plenty of source of information online so, I won't repeat the whole thing here. In very short and very simplified, talking about humans...

Ploidy, meiosis and fecundation

Both parents are diploid (2N=46). Both parents have specific cells that are undergoing meiosis (in testis and in ovaries). After meiosis, the cells (spermatozoids and ovules) are haploid (1N = 23). The spermatozoid and the ovule fuse (fecundation) to make another diploid (2N = 46) individual. This diploid individual will grow for about 9 months in the mother's uterus before coming out!

Crossover

Now, during meiosis, cross-over occurs. It is a mechanism where the two chromosomes mix up their material. When the cross-over occurs in the father for example, it is the chromosome that the father receive from his father and the chromosome that he received from his mother that mix-up their genetic material.

Possible chromosomes

Let's say the maternally inherited chromosome of the dad is

AAAAAAAAAAAAAA


Let's say the paternally inherited chromosome of the dad is

BBBBBBBBBBBBBB


Then, in the meiosis (where spermatozoids are produced), the two chromosome might mix up and the haploid spermatzoid can end up with something like

BBBBBBBBBBBAAA


Now, on the mother's side, her two chromosomes are, say,

CCCCCCCCCCCCCC


and 

DDDDDDDDDDDDDD


The mother might therefore transmit

DDDDDDDCCCCCCC


The (diploid) baby would then be 

BBBBBBBBBBBAAA
DDDDDDDCCCCCCC


So of course, the chromosomes present in the baby are a bit different form the ones present in the parents but we can still recognize them. If there is doubt who the father is, a genetic analysis of another father might return the chromosomes

BBBBBBBBBBBBBB
FFFFFFFFFFFFFF


, then we would know this individual cannot be the father because he could have not transmitted the AAA part of the chromosome found in the kid.