Not to condone the idea of being dishonest, but everyone should understand what does and does not come back during a PRIA records search.
If you are (as it sounds) someone who has never worked for a 121 or 135 air carrier before (or within the last 5 years), the only part of PRIA that will apply to you is what your new airline will get from the FAA.
All they will get from the FAA is a one page letter showing your current certificates (i.e. Commercial Pilot AMEL ASEL), CFI certs (if you hold any), A&P certs (again, if you hold any), medical class and date, and whether you either have violations or suspensions, or are currently under investigation. That's it.
Knowing that some pilots actually understand the above, I have heard that some regionals are asking applications to fill out a Freedom of Information (FOIA) request, which will allow the FAA to send them copies of EVERY record that they have on you. This will include written test results, 8710s, actual checkride results (disapproval, discontinuances, etc) from Part 61 training events, and I'm not quite sure what shows up in this process from 141/142. Given the fact that "stage checks" under 141 are not considered checkrides (at least by the FAA), my assumption here would be that all that anyone gets from a 141/142 school would be a "graduation completion" type of document, stating that you finished their program with a Commercial pilot cert.
While we can all agree that this industry and profession is about honesty and integrity, in this day and age where everyone worries about liability (no one more so than airline HR departments) and bad PR issues (which has virtually come to the point of "OMG!!!! You farted sideways in your sleep once?!??!! We can't hire you because of that!!!!!!!), the flip side of this, is that if someone is never going to find out about something that you did, which has zero basis on on your ability to perform an essential job function, 13 years ago, one can argue that it's not entirely in your best interest to tell them.
In the end, do what you feel is right.