Good differential, here's my two cents,
Mumps I would think would be a long shot since most children get MMR or MMRV vaccine unless you know this child never received said vaccine. No drainage indicated if this is the case, Tylenol and just let the virus run its course with supportive measures if necessary.
Buccal space infection is a tough push without some sort of identifiable etiology (carious tooth that would reach this space, recent deep tissue masticatory trauma? - r/o with detailed history?) Also, these types of infections are pretty rare as is and in a child this young, I would think even more rare as there are not likely any odontogenic sources that can reach below/above the buccinator muscle attachment at the age of 6 (1st molar roots are not likely fully developed), but I have seen it.... if this continues to be high in the differential, you can drain intraorally but be mindful of the anatomy and make sure you know what you are draining.
Acute sialadenitis of the parotid is probably what I would bet on with the reported symptom of pain on salivation and clinical picture. This could be on account of masticatory trauma to stenson's duct with scarring and fibrosis that leads to decrease in salivary flow and eventual bacterial influx. I would imaging pen covering this fine since this is early on, if I had my pick, amp or unasyn but cost becomes an issue, flagyl is a nice way to augment the pen, but I would not think this acute 1 day cellulitis to be grossly anaerobic at this point. Some would argue surgical drainage is the way to go, but if this were my child, I would wait to see how he fairs on the IV abx and drain only if there is a suspected abscess consolidation and get a CT to guide you/or who ever this gets referred to. Remember the facial n. branches within the substance of the parotid and damaging that in a child (or anyone) would be a serious bummer...
I have a feeling there is more to this story with the previous mysterious admission, make sure to r/o any existing medical conditions (autoimmune?, immunocompromised? possible salivary gland neoplasm that is not otherwise clinically significant at this time?). I would not expect the mcs to be very informative, will likely return oral flora which you are already empirically treating.
This is a good case to discuss, thanks. I am sure I haven't covered everything and have missed something, but that's all I got for now.