Side-effects are extremely rare - far less than other medicines. These are utterly dwarfed by the effects of the infectious diseases they prevent.
In the rare cases where the child may have a pre-existing condition that could be aggravated by immunisations, doctors are able to provide medical exemptions for diseases. In the US, approximately 0.05% of children enrolled in schools are exempt from immunisations on medical grounds.
If parents ruled out immunisations and medical procedures that had the possibility of aggravating an undiagnosed illness, they would deny their child almost all medical treatment. Almost all treatments carry the risk of aggravating an undiagnosed illness in some form. We must devise a standard for "safety" that means that the risk of aggravating an underlying illness or disease is sufficiently small to justify the widespread use of the treatment. Vaccines would meet this standard for safety, even if it cannot be called 100% safe.