Is it normal for the secondary air injection pump to briefly turn on after driving the car, then putting it in park?

 

No.
It is only normal for it to run with a cold engine start for 30-45 seconds.

 

Beg to differ, mine runs on occasion after slowing to a stop (say at a light). Runs for a few then shuts off. I believe it is cycled by the emissions systems.

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Well, then I don't understand the function. What I read was that it provided extra air (oxygen) in the exhaust to cause the cats to react more intensely, in order to heat them up faster.

 

Yea Mark, that's the purpose on any car that has a secondary pump, heat up the cats quicker on a cold start so it can go closed loop quicker and therefore cut down on overall emissions.
At least, that's what I learned in shop class back in 1961 when they were dreaming about the days when we could go all digital on engine control and stop greenhouse effect. ( NOT )

 

it can be normal. the pcm tests the secondary air injection system once per drive cycle. it needs engine coolant over 122F, engine at idle with a/c off. it's possible one of those conditions isn't met during your drive cycle (i'd assume engine at idle).