Can I get a work permit while waiting for F2B petition to be approved?

What country are you from?

No, you cannot get any immigration benefits based on your father's I-130 until a visa priority date becomes available. If you will be traveling, it would have to be on the basis of your F-1. However, you should know that F-1 is a nonimmigrant intent visa and if you have a pending I-130 and travel with F-1 visa, you may not be permitted to enter the US as you will be seen as having immigrant intent which is not permitted with F-1 status.

Hello: The visa number must be current and you must be able to adjust to get a work permit. It will take many years.

You should sit down with a qualified immigration lawyer as there are no simple answers here.

F2B is backed up 8.5 years (11 years for Philippines, 20 years for Mexico) Your father filing of a family petition for you does not give you any immigration benefits, no work authorization, no permission to remain in the U.S., nothing. You could travel while you are on a valid F-1, but I would advise against it because, once your father files the petition for you, it will act as evidence of your intent to immigrate into the U.S., and USCIS holds that such intent is incompatible with a non-immigrant visa (including F-1). So you might be denied admission when trying to re-enter after a trip abroad. Filing of a family petition will also hurt your chances of getting any other non-immigrant visa, such as H-1B, for the same reason ("demonstrated immigrant intent") I suggest that you should get an immigration attorney to review your father's case: depending on the details, you might be eligible to receive a green card as a beneficiary of your father's adjustment application.

You can only get a work permit after the I-485 is filed, and for that to be filed your priority date would need to be current. That is probably unlikely to happen before your studies are finished. Hopefully you will be able to secure an H-1b to cover the gap time.

The time frame for an F2B depends on the country you are immigrating from. Some countries, like India, take longer. You can check on the Department of State's website for priority dates that they are currently working on.