The "airport officials" won't care they have no authority over allowing you to travel. One of the biggest issues today is Child Trafficking, and many immigration officials have become very strict on making sure that the minor is visiting for legitimate purposes.
Any issues you may have would be determined by the Country(ies) you are visiting and the Country you are from. The purpose of the visit would also play into this. That is they may give you a little more leeway if you were on a school trip or other organized educational trip compared to just visiting "friends".
So it is very possible that if you only have permission from 1 parent, you could be denied entry. If the reason you can'[t get your other parent's permission is just logistics, such as they are unavailable, if you have paperwork showing why you can't get their permission would help.
If they just don't want you to go not only could that be an issue, but if the divorce decree gives both parents equal legal custody they could actually keep you from going. If you did go and one parent allowed it that parent could actually be in legal trouble for violation of the agreement.
Now, because of various immigration requirements the airlines have a responsibility to verify that your paperwork is in order. As a result if you don't have permission from both parents due to these issues they may not even allow you on the flight.