Those plants don't cause problems, since the pollen stays put, either in the flower, or on the insects. The pollens that do cause allergies are from plants which rely on the wind to distribute the tiny granules from one place to another. These microscopic particles are produced by plants which have flowers that are not noticeable, either visually or by scent, such as the cottonwood or sycamore. Well, for several months of the year, every time you take a breath, you're breathing in pollen. Even if you manage to get through the spring without an allergic attack, you still need to survive the blooming of the grasses, the weeds, and the sagebrush which bloom all summer, as well as the mold and mildew spores which are around all year long.
Rain helps, some.
It washes the pollen out of the air, and weighs it down so it can't blow around. But it also stimulates the growth of mold and mildew, and encourages weeds to flourish. So instead of sneezing at one thing, you're sneezing at another. Mostly, it's genetics, which means you can blame your mother and father. On a more direct level, if you are allergic to something that means that your body produces a molecule called lgE, which is a kind of an antibody, related to those molecules which fight off infection.