Question:

Can an employer refuse to allow somone going out on FMLA for a sick spouse or child to use their sick time?

Response:

The Family Medical Leave Act was established to provide employees the opportunity to take leave for their own serious medical condition or to care for a family member who has a serious medical condition.  Most employers will require that employees use their sick and vacation time while they are on FMLA, so that employees don’t have opportunities to take more time off after they have already taken 12 weeks.
If you are requesting FMLA for basic sick children or a sick spouse, the department of labor has established a listing of reasons to take leave, and they have definitions of a serious medical condition, your denial to take leave may depend on the seriousness of your medical condition.