Give parts 2 and 3 of your P45 directly to your new employer. If you are claiming Jobseeker's Allowance or Employment and Support allowance, give parts 2 and 3 to the Jobcentre. Do not send it in to HMRC, which already has a record of it.
The P45 comes in four parts: Part 1, Part 1a, Part 2, and Part 3. According to HM Revenue and Customs, your employer is responsible to send information for Part 1 of the form to HRMC and to give Parts 1a, 2 and 3 to you. Part 1a is intended for your own personal reference. The P45 form is a record of your earnings and shows the HMRC and your future employer or Jobcentre how much tax has been deducted by that point in the year.