In plain English
FAR 52.232-3, Payments under Personal Services Contracts, is a contract clause prescribed at FAR 32.111(a)(3), most recently dated Apr 1984. The complete official text is reproduced below, verbatim, from GSA's published FAR source files.
Its text contains no sentence requiring insertion into subcontracts; see the flowdown section below for what that does and does not mean.
Does it flow down to subcontracts?
No flowdown mandate found in the clause text
We scanned the full clause text and found no sentence directing the contractor to insert this clause into subcontracts. That is a statement about the text, not legal advice: a prime contractor may still flow terms down contractually, an agency supplement (DFARS, VAAR, …) may add requirements, and clauses listed below (if any) may order this clause into subcontracts from the outside.
Where it's prescribed
As prescribed in 32.111(a)(3), insert the following clause, appropriately modified with respect to payment due dates in accordance with agency regulations, in solicitations and contracts for personal services:
Prescribing reference: FAR 32.111(a)(3).
The official text, verbatim
As prescribed in 32.111(a)(3), insert the following clause, appropriately modified with respect to payment due dates in accordance with agency regulations, in solicitations and contracts for personal services:
Payments under Personal Services Contracts (Apr 1984)
The Government shall pay the Contractor for the services performed by the Contractor, as set forth in the Schedule of this contract, at the rates prescribed, upon the submission by the Contractor of proper invoices or time statements to the office or officer designated and at the time provided for in this contract. The Government shall also pay the Contractor-
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(a) A per diem rate in lieu of subsistence for each day the Contractor is in a travel status away from home or regular place of employment in accordance with Federal Travel Regulations (41 CFR 101-7) as authorized in appropriate Travel Orders; and
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(b) Any other transportation expenses if provided for in the Schedule.
(End of clause)