Parasites do not cause asthma. Asthma results from adrenal dysfunction. The reason children will generally outgrow their allergies by the age of 5 is it sometimes takes a little more time for their adrenal glands to mature. If the child is on steroids though this can atrophy the adrenals making the underlying condition worse by preventing the adrenals from doing their job.
Asthma itself is a form of allergy. It is an inflammatory allergic response.