Saliva is produced by six major salivary glands, three each side. Parotid, in your cheeks. Submandibular, under the lower jaw. Sublingual, under the tongue. Plus a whole bunch of minor salivary glands lining the cheeks. We produce and swallow it all the time, without realising.
Archie needs it suctioning for two reasons. Firstly, he no longer has an effective swallow, so is at risk of drowning in his own saliva. Secondly, his lilps aren't sealing together because the lip muscles are passive, so saliva will just flow out the front of his mouth. The first is the most risky, obviously.
It's not produced in the brain.
I suspect Dance has mixed up 'parotid' and 'pituitary'.