Head lice and body lice are extremely close relatives, usually treated as forms of Pediculus humanus. The body louse differs mainly in habit: it lives and lays in the seams of clothing and visits the skin to feed. That habitat did not exist until humans wore clothes, so the divergence of the body louse from the head louse cannot predate clothing.
Molecular clock estimates published from 2003 onwards placed that divergence roughly between 70,000 and 170,000 years ago, depending on the genetic markers, calibration and demographic model used. Textiles themselves almost never survive from that period, so the louse offers evidence where archaeology is silent.
The method's weakness is the clock. Molecular dates carry wide confidence intervals and shift with each recalibration, so the figures should be read as a broad window rather than a date. Two facts that do not depend on the clock: head lice do not jump or fly, and infestation is unrelated to cleanliness.