Can my employer class dependency leave as a sickness leave on my record at work?

Yes... unless you used your vacation time... you wouldn't get extra paid time off for that... did you use FMLA and take time without pay? But you would typically go through your sick leave first... then your vacation in order to keep a paycheck coming during FMLA. If your time out exceeded vacation and sick leave then the remainder would be unpaid.

Agree, you will use your vacation and then sick leave first, if that is all gone, you would apply for FMLA ( but that is not one day here and there but long term)