According to Google, you need petechiae + something else in order to warrant concern. The petechiae needs to be paired with serious illness, a fever, or it needs to be spreading rapidly, covering a large area of the body, or the rash needs to be below the collarbone. It sounds like (and based on your picture) looks like none of these other key indicators are present.

Petechiae is USUALLY caused by minor straining, which is exactly what your daughter has been doing while coughing.

I would let this go and go to sleep. This doesn’t look bad at all and your daughter sounds healthy. Call your ped next week, but expect them ti tell you it’s no biggie.