When grass is cut, it is understandably a huge trauma to the grass. It responds to this trauma by releasing volatile organic compounds, which cause the fresh cut grass smell. Volatile organic compounds are carbon-based molecules that cause very strong smells. The odor is really a distress signal by the grass causing it to heal its wounds faster or prevent bacterial and fungal growth.

A particularly interesting purpose of the chemicals is that they attract big bugs to prey upon the caterpillars that attack the grass. The allergy likely has nothing to do with this smell.

Grass allergies are generally towards grass pollen. When the grass is mowed, it kicks up all of the pollen into the air which causes the allergy.

The pollen going to the air is a side effect of grass being cut, but has nothing to do with the smell of the grass itself. In fact, as grasses get taller, they release more pollen, so not mowing the grass could make allergies worse.

It is unlikely that chlorophyll has a significant odor, especially when compared to the volatile organic compounds released by the plant. The class of organic molecules that chlorophyll belongs to are not generally recognized as having strong smells.