Do I need the father's permission to take child on a family vacation to Peru

Generally speaking, you will not need the permission of the court to take the children on a vacation in or out of the country as long as yourself and the children have not been ordered by the court to not leave the jurisdiction. As you note in your question, the father has visitation rights with the children so he is entitled to one weekend a month but the question is which weekend and how does the designated weekend get set. That being said, as long as you are not violating his rights to visitation there should not be a problem. It may be best however, for you to notify the father of your plans and provide him with an itinerary for your flights and hotels etc. You do not want him to think that you have left for an indefinite period of time or have decided to relocate. If you are planning on relocation of any sort, you will need his permission or that of the court.

As is typical of a lawyer, there are more questions to be asked. Does the child have a passport? You may find that both parents need to sign off on the passport. It is also my experience that some air lines require a form indicating the non-custodial parent has given permission.

I practice in Washington and have practiced in California. Both states distinguish between physical and legal custody. Washington uses "residential time" instead of physical custody. So you are the primary residential parent. The parenting plan also designates decision making, which can be joint or sole.

You do not need court permission to take the child out of the country. (Hawaii is in the U.S.) You may need father's permission. Check with your air carrier if they require a form.

Although I do not practice in your state, if you get his permission I don't think that you would need to get the court involved. You should contact a local attorney.