Tax query: two jobs

Hello, Recently received a P800 underpayment notice from HMRC and I'm needing some guidance. I have two jobs with different employers: one part-time (7.5 hours a week) and one full-time (37.5 hours). The part-time job was previously full-time; however, I reduced my hours to part-time to take up the new full-time position. Having checked, my tax code for my old full-time position was never changed when I switched to part-time hours, resulting in a part-time and full-time job with the same tax code. Am I liable for the underpayment, or is my employer? Any information would be greatly appreciated.

Hi,
You are liable for the underpayment, PAYE is Income Tax which is deducted and paid to HMRC on your behalf by your Employer.
The other way to pay Income Tax is Self-Assessment.  
It is your responsibilty to check your tax coding is correct.
You can find more information here: Tax codes
Thank you.