It is my understanding this is now or will soon be a moot point as
P20W student data collection is expanding.
As the SAT has been revised and is now described as a predictor of college and career readiness and success, it matches the broad data descriptor #10 from the US DoE data collection factsheet dated 2009:
Data on whether K-12 students are prepared to succeed in college. The
factsheet also states this requirement: "The ability to share data from preschool through postsecondary education data systems."
This indicates that SAT scores would become part of the "comprehensive data systems... will capture data on students from one grade to the next", sharing the data with governments, researchers, and postsecondary institutions.
Regardless of whether parents decide to keep or strike SAT scores, these scores will already be recorded in other databases.
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The US DoE's 2009 data collection factsheet is archived on the WayBack Machine - https://web.archive.org/web/20210809095250/https://www2.ed.gov/programs/slds/factsheet.html