Hi Janet,
I am currently helping prepare my parents in law application for an onshore contributory aged parent visa, so by no means an expert but they are in the same boat.
Your father (assuming he meets the other requirements of the visa BoF etc) should be able to submit his onshore application whilst he is here next month, as long as his current visa allows him to submit further applications and he doesnt have a no further stay condition applied to his visa . It seems strange that this is allowed I agree but if you think about there is no other way anyone would realistically ever be able to apply for this visa.
If he applies offshore he is as far as I am aware able to apply for other visas to visit but I havent looked in to this in detail but I know many people visit whilst waiting for their applications. Hopefully some other forum members can comment on this.
In terms of the bridging visa this comes into place when his current visa expires, so three months after landing in Oz or 12 months after grant (allowing for multiple 3 month visits). As far as I understand it he can come and go on his visitor visa (whilst it is still valid) once his onshore application is submitted. I would just make sure that he is in Oz when the visitor visa expires, so 12 months after grant (assuming it hasnt already expired due to a visit of more than 3 months), allowing him to transition onto a bridging visa. The other thing I think you would need to be mindful of is that he needs to be in Oz when the visa is granted which according to IMMI are currently taking about 6 months to process.
This is just my understanding and hopefully some other forum members comment or as suggested above you contact an agent.
Good luck!