I'm majoring in Ecology & Evolutionary Biology, Microbiology, and Bioinformatics. I have a textbook about gene sequencing right next to me, coincidentally, so I hope I can help out a bit.
DNA sequencing is not a go-to for identification of eukaryotic organisms (or for most pathogens at all). But if your parasite has garnered government attention, CDC would get on identification very quickly with state of the art high-throughput sequencing.
High-througput DNA sequencing is fast. A whole human genome can be assembled with automatic programs (which could take a day) and additional human analysis (which could take 3-4 days). So that's around 4-5 days with leading sequencer technology.
DNA sequencing that is used to identify organisms doesn't assemble a genome. DNA fragments are compared with various genome databases to identify the organism, or what it is most closely related to, which is a much faster process (~6 hours?).
Having a whole genome doesn't tell you much once you've assembled it, especially if there are no similar organisms in the genome databanks, so for identification purposes, you don't need to assemble a genome.
If it's from another country and that deadly, it's very likely to make world news and the CDC would already be tracking it asap--especially if the US is close to this country.
Additional opinions, feel free to disregard if you've got your critter all figured out:
A parasite is an organism that relies on a host to survive and can often go undetected and cause very little harm to its host. Often it is in the best interest of the parasite not to kill its host, because it needs the host.
A parasitoid is an organism that has a parasitic relationship with its host but specifically results in death of the host. Usually that indicates the parasitic organism no longer needs a host to reproduce at that stage of its life cycle.
A pathogen is a disease-causing organism that causes notable damage to cells, but the term isn't usually applied to macroscopic things.
I would suggest basing it off of an existing organism--this will save you grief of researching the plausibility of various aspects of your critter, and also amplify the creep-out factor. There are some super terrifying weird things out there, like the parastic screw worm fly, which was eradicated from the US--but came back!
Also, it would be incredibly bizarre to suddenly have a eukaryotic parasite outbreak if it's spreading at the rate that it is. Deadly emerging diseases are usually viruses or bacteria because these can mutate much faster and produce new strains, unlike eukaryotic organisms. The reason many of these have high mortality is because they are relatively new strains that have not before infected humans. This is a trend seen with infectious diseases, and usually levels out in mortality after a few years because, again, it is often in the pathogen's best interest to keep its host alive.
I enjoy worldbuilding my own pathogens.