I am currently having an F1 Visa which will expire this December and I applied for PERM which is still pending. Can I travel abroad and come back to US without affecting my pending perm application?
Short answer - NO, not without first getting Advance Parole.
DO NOT leave the U.S. without the Advanced Parole Document!
If you do, your application will be considered abandoned & you just wasted a bunch of time and money.
Since you didn't apply for this along with the Adjustment of Status application, it is a separate process (Form I-131) with a separate $360 fee. Depending how long ago you submitted your application, it might be better to just wait for the Green Card.
If you really need to travel (medical or family emergency, etc), apply for the travel document as soon as possible & hopefully it'll be approved quickly. You need to have the paper with you when you try to re-enter the country.
If you have applied for an adjustment of status and your application is still pending you can apply for Advance Parole Document from USCIS in which allows you to travel outside United States without having your application for adjustment of status be deemed abandoned by the USCIS.
PERM is done with regards to the job opening and is a Department of Labor process. Until your application to adjust status reaches USCIS (i.e.: you file form I-485), you're in the same status as you were before and can travel (assuming you have a valid visa to be able to come back) as you like.
Once you file I-485, leaving the country without the "Advance Parole" is considered abandoning the application.
PERM is not related to your status.
In website it says:
Q: Am I allowed to leave the U.S. while my PERM labor certification application is pending?
A: Yes, you can. However, note that just because your PERM labor certification application is filed and pending will not aid your re-entry into the U.S. You still will have to secure a visa for re-entry and resolve all related issues. To this end, prior to your departure, consult a seasoned immigration attorney who’s able to resolve all issues that could impact your re-entry.