It is unlawful for any student to enroll in school unless he has been immunized, as required under the rules and regulations of the health services division [public health division] of the health and environment department [department of health], and can provide satisfactory evidence of such immunization. Provided that, if he produces satisfactory evidence of having begun the process of immunization, he may enroll and attend school as long as the immunization process is being accomplished in the prescribed manner. It is unlawful for any parent to refuse or neglect to have his child immunized, as required by this section, unless the child is properly exempted.