Our bylaws state that if any one person is nominated for a position, then an election does not have to take place. The recording secretary casts the lone vote to elect them into office. Therefore, your secretary that is now nominated for President would have been voted in if this situation was in our school.
Is your current president willing to stay on? If you have terms listed in your bylaws, then this situation should be covered as well. Refer to Roberts Rules if it states that the bylaws are based on them. However, you should also propose to update bylaws to clarify this for next year.
Personally, depending on the reason for the resignations, I wouldn't want to take a chance on electing either in since they could not fulfill their old positions as Treasurer or Secretary, how do they think they would fill these new positions? Something for your members to consider before casting their votes!

First you said your bylaws make no reference to if there is no nomination for a officer what you would do, is there a term limit? Is your president willing to do it another year? However if the S gets on the ballot she is pretty much a done deal, if she is the only nominee, unless you have a majority type thing going on, I mean on a ballot you either vote for one of the choices, or don;t vote for that position, one vote she could be in. I am not aware of Roberts Rules so this is my insight. Heck each year I run against myself for president...

My main concern is why they are back wanting to do another job when they gave up one, do they really think the president job is easier than the secretary job. Why do I feel there is some negativity involved with this and that they are trying to take control of your group. Please tell us more...I feel like there is more to this that meets the eye.

I'm with Silver. Finding out the reasons behind the resignations would be valuable in determining whether these parents are worth voting for or not. If there is a pattern of walking away from things they've volunteered to do that's one thing, but maybe there were some issues within the board that they found stressful and difficult and they finally just gave up. They could be stepping up to try to change what they found wrong with the group. Ultimately it will be up to your membership to make the decision about whether they deserve the office or not.

I also recommend going to Robert's Rules of Order for how to handle this potential situation, but don't make a big deal out of it unless it happens.

What's the root of the problem with the resignations? Perhaps once the reason why they resigned is out others would feel more confident in their nominations for office.
If your by-laws state ROR follow ROR. If not perhaps your current President would be willing to fill in until a replacement is found?

We have had an interesting year within our PTO. In Jan. our T & S both resigned and left our current P & VP to finish up with all PTO business. After asking several members and a few months later someone took on both offices. We just had nominations for our 07-08 officers and low and behold the 2 resigning officers have nominated themselfs for the offices. The resigning T was nominated for 3 offices T, S & VP and the resigning S was nominated for the P. There are several members that are reluctant on voting them into office after they currently were elected then after a few months residned from the offices that they were voted into. There are 2 nominations per office except the P. What happens if the nominee is not voted into the P office? Does the current P stay in office? I would think that the newly elected VP would fill in until a P is elected. There are no details on this matter in the Bylaws. I want to know from the experts as to what may or may not happen.

Thank you in advance,