How long does foreign DNA stay intact in human blood?
The De Vlaminck Lab has extensively studied the origins of circulating cell-free DNA (cfDNA) in human plasma and its utility in detecting infections and organ injury.1,2  Concerning your question of parasite DNA detection and parsing signal from noise, I don't think you'll find a better resource than their 2020 publication in Microbiome.3
If the (uncited) claims on their website are to be believed, cfDNA from all sources is cleared within one hour after release into the blood.

Our research pursues technologies and applications of cell-free DNA in infectious and immune-related disease. The great promise of cfDNA in diagnostic medicine derives from i) its abundance: 10-100 billion molecules of cell-free DNA can be isolated from just 1 mL of plasma. The combined DNA sequence of these molecules is sufficient to cover the human genome 1,000 to 10,000 fold. ii) Its origin: cell-free DNA is derived from dead cells and comprises rich information about cells in the blood and any vascularized tissue that can be accessed noninvasively. iii) Its short lifetime: cell-free DNA is cleared from the blood within 60 minutes. Cell-free DNA therefore provides a very dynamic window into health.


References

Burnham P, Kim MS, Agbor-Enoh S, Luikart H, Valantine HA, Khush KK, De Vlaminck I. Single-stranded DNA library preparation uncovers the origin and diversity of ultrashort cell-free DNA in plasma. Sci Rep. 2016 Jun 14;6:27859.
Cheng AP, Burnham P, Lee JR, Cheng MP, Suthanthiran M, Dadhania D, De Vlaminck I. A cell-free DNA metagenomic sequencing assay that integrates the host injury response to infection. Proc Natl Acad Sci U S A. 2019 Sep 10;116(37):18738-18744.
Burnham P, Gomez-Lopez N, Heyang M, Cheng AP, Lenz JS, Dadhania DM, Lee JR, Suthanthiran M, Romero R, De Vlaminck I. Separating the signal from the noise in metagenomic cell-free DNA sequencing. Microbiome. 2020 Feb 11;8(1):18.