What to call the collective parts of a day?

I would say that the periods of a day are simply time periods. For example, the definition of morning time says it is:

the time period between dawn and noon

You could also call them periods. The definition of evening says:

1. The period of decreasing daylight between afternoon and night.
2. The period between sunset or the evening meal and bedtime: a quiet evening at home.
3. A later period or time: in the evening of one's life.

Edit: A more specialized term, used in broadcast programming, is dayparting. That is,

In Broadcast programming, dayparting is the practice of dividing the day into several parts, during each of which a different type of radio programming or television programming apropos for that time is aired. Television programs are most often geared toward a particular demographic, and what the target audience typically engages in at that time. Arbitron, the leading audience measurement ratings service in the United States, divides a weekday into five dayparts: morning drive time (6-10 am), midday (10 am-3 pm), afternoon drive (3-7 pm), evenings (7 pm-12 midnight), and overnight (midnight-6 am; Arbitron generally does not measure during this time period).

You could adopt this term for the use you seek, although many people would not know its precise origin. The individual parts of the day could be termed dayparts.

"Dayparting" is probably the correct technical term, but it's not one I've come across before. More commonly, these are simply "times of day".

They are of course often used as greetings "[Good] Morning" / "[Good] Evening" etc., and the common expression for someone being rude enough not to exchange such a greeting is "[he/she] wouldn't give me the time of day".

Admittedly the "time of day" could also be interpreted as a reference to the exact time - but that is more commonly just referred to as "time".