DNA replication and combination > "Each gamete is genetically unique because the DNA of the parent cell is shuffled before the cell divides. This helps ensure that the new organisms formed as a result of sexual reproduction are also unique." Then why do we say that the DNA of the parent influences the characteristics of the child while the DNA of the child is formed as a combination of shuffled up nitrogenous and phosphate bases?

To explain it briefly:

Lets take a human as example, you are diploid and you have a pair of 23 chromosomes (= total 46) and the sex chromosomes which I will exclude for this explanation.

Your gamete is haploid and has therefore only one of the two paired chromosomes. So for every chromosome pair there are 2 possible chromosomes. So in total you have 223 possibilites to arrange chromosomes. The gamete from the sexual mating partner also has 223 possibilites to arrange chromosomes. So in total a new diploid organism has (223)x(223) possibilites. So unique.

However, it is always the chromosomes of the parents. So yes, parents influence the children because the genetic information of the children can be found in one of the two parents.

Furthermore you have recombination, but you already asked another question about this. So I will not expand this answer.

There are relevant articles in Wikipedia on genetic recombination and meiosis that I would recommend.