Can I secretly record my aging father to show his growing mental decline for future reference?

Secretly, it is an invasion of privacy, legally, imo.

If you were using it as a documentary, you would need written, legal permission.
If later, you are making it into a movie, you would need a contract.
If ever, you posted it on social media and it became viral, a family member objected and sued you, that would be a hot mess.
Memories are enough, and the mind has a way of softening or rewriting those.
Give the privacy and respect to your father that any human being should get.
Please consider his needs above your own in this instance.

Hoping that your motivation is not coming from a need to be heard, to be understood, or to prove to siblings (or someone else) that he is, indeed declining.
You can tell that to the caregivers here who are going through what you are, as many can identify with you, and will understand.
Are the doctors listening to you?

They might be useful for showing her Dr, but no one else. Keeping videos of LO's, for the sake of the memories of good times past is a great idea,  and I wish I had done that more. Snapshot pix are great, but those videos are priceless!

They will be nice to view, when you wish to remember them BEFORE Dementia robbed them of their faculties. 

There comes a time, after the've passed away,  when you finally begin remembering them Before illness,  and that is now and finally how I remember my folks, despite the horrible endings to their lives! The videos will be great for that reason, the happy times! 

I think Grammyteacher hits the nail on the head - why the need for secrecy? Why not record conversations and events in any case, for posterity even if you like; and if they also serve as a useful baseline for comparison there's nothing wrong with that.

I have short video clips over the course of time and progression. I never did this in sneaky ways and I have never showed them to her or anyone other than my husband (and happy ones to close family) when we are trying to track her progress. Even though we keep a journal, it is easy to not realize how much decline has happened. 

Some of the videos are actually sharable moments too...her singing Happy birthday...her smiling and laughing as she tells us stories (even though you can't understand the story)...her holding her great grandbaby when he was new...who is 3 today...sad thing is she doesn't even "see" the newest great grandbaby that is 5 months old...progression. Those are times when the progression really stands out. Without the video, I am not sure I  would have as realistic can memory of how she was 3 years ago. For me, it all just smooshes together.

I think taking video to track progress or for remembering sweet moments, is fine....if you are trying to "prove" something about progression or are going to use it in a way that could be degrading or disrespectful to the person, that is wrong.

I tape recorded my mom a couple times because twisted sisters and auntie dearest did not believe how sick mom had become with her Alzheimer's. Did it help? No not really. Some just prefer their state of denial and that is nearly impossible to change.

Is there some reason why you feel the need to record him? When my mother started showing signs of decline, there was no doubt in anyone's mind that she was in the first stages of dementia. I didn't have to prove it to anyone. I would suggest having a meeting with sibs if you have any and sharing what you've observed. You can also go to his doctor and privately share your concerns. The doctor should order psychological testing which can prove or disprove your suspicions to anyone. If you should want to use these secret recordings, I am not certain they would be admissible as evidence since they were made without his knowledge. It could be considered an invasion of his privacy.

You could do that but it will serve no purpose other than to hurt/anger him.

If you are dealing with dementia and short term memory loss he might recognise his decline in that moment by viewing some video but would most likely forget and/or be in denial about it. You'd have to keep repeating the process.

This seems to me to be a punitive approach. Don't do it.