Part-time employees have exactly the same entitlements as full- time employees. In order to qualify for statutory sick pay your employee has to earn above the lower earnings limit.
They will qualify for statutory sick pay if they cannot work due to sickness for four or more days in a row. These are calendar days, not working days. They will not receive statutory sick pay for the first three qualifying days of sickness but will receive it for any ongoing period of sickness after that time. The daily rate of statutory sick pay is worked out by dividing the weekly rate by the number of days this employee works per week.
If you offer contractual sick pay then your part-time employees will qualify in the same way that your full time employees would. You must make sure that you treat your part timer the same way as you would treat a full-time employee.