When should a parent be concerned about a fever for their child?

Depends on age. Any fever in an infant under 2 months is cause for concern. It is difficult to distinguish serious illness such as bacteria in the blood (bacteremia) or meningitis from non-serious illnesses such as viral infections in children in this age. Thus any child under 2 months with fever needs to be evaluated in an office or even an er setting with blood, urine, and spinal fluid testing.
Sick-looking. Wen it comes to fever, it is not necessarily the height of the fever that should be of concern. If the child has a fever of 103 and is still playing and eating well, you can be less concerned than if she had a fever of 101 but has no energy, wimpers, and moans and has no appetite. In that case, you should call the pediatrician and arrange for an appointment the same day.
Depends on age. Most children with fevers over 2 months of age will have minor viral illnesses or bacterial infections such as ear infections that are not a cause for concern. However, any fever combined with being very ill appearing, or fevers that do not improve with tylenol, (acetaminophen) should be assessed by a pediatrician.
Depends on age. Any infant under 8 weeks of age who has a fever above 100.6 rectally should at least be seen by her pediatrician. With a significant fever, it is often standard practice to have these infants admitted to the hospital and tests performed to make sure they don't have a life-threatening bacterial infection before they can be discharged.
Depends on age. Parents shouldn't be concerned for most fevers. However, any fever in a child two months or younger should be reported to a physician, because they are more susceptible to serious infections and may need a physical exam and have laboratory samples drawn to explore the fever further. In older children, parents should be more concerned with how the child is acting than how high the fever gets.