Since most phenotypic characters and biochemical pathways in any living organism are based on its genetic background, is the updating of parasite's genome happens in response to the host genome evolution?
your question sounds very broad and general. Most of coevolution between plant and pathogen happens in pathways related to defense and anti-defense mechanisms. Half of the answer you can find in recent work devoted to Solanaceae: Moghe GD, Leong BJ, Hurney SM, Daniel Jones A, Last RL. Evolutionary routes to biochemical innovation revealed by integrative analysis of a plant-defense related specialized metabolic pathway. Elife. 2017 Aug 30;6. pii: e28468. doi: 10.7554/eLife.28468.
It depends on the type of relationship between the two individuals. In the case of symbiosis, yes there is a common evolution between the individuals, but if they are parasitic then the evolution is not tolerated, still it is necessary to see what parasitism it is.
This is true. In case of obligate pathogens there is a reduction in number of functional genes as the pathogen depends on the host more and more over time (Example: Mycobacterium leprae). Obligate endosymbionts have shown extensive genome reduction. Please read the following papers.
The decaying genome of Mycobacterium leprae (Article The decaying genome of Mycobacterium leprae
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Extreme genome reduction in symbiotic bacteria (https://www.nature.com/articles/nrmicro2670)
An evolutionary analysis of genome expansion and pathogenicity in Escherichia coli (Article An evolutionary analysis of genome expansion and pathogenici...
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Contrasting host–pathogen interactions and genome evolution in two generalist and specialist microsporidian pathogens of mosquitoes (https://www.nature.com/articles/ncomms8121)