"Could this be them?"

From the OP...

"...William Henry Armstrong, death certificate...states that William's parents were named James and Mary..."

As anyone with some experience in genealogy research will vouch for, death certificate accuracy relative to parental names is generally predicated upon the accuracy of the informant's knowledge. It seems unlikly that, among 4 siblings or a spouse, the paternal name might have been mis-recorded as "James" here...but it certainly is an interesting plausibility.

Is there an informant name recorded in William's death certificate?

Also plausible (but somewhat grasping at straws) is a given name of "John James". In Irish tradition (IIRC), isn't the middle name often the preferred in common use?