In general, you cannot treat part-time workers worse than full-time workers. Your absence policy should apply the same rules for all employees.
In terms of sick pay, part-timers qualify for statutory sick pay in much the same way as full-time employees. So they are entitled to SSP from the fourth day of illness, but only for days when they would otherwise be at work. If your business offers a more generous sick pay scheme, this should also apply to part-time workers.
Similar rules apply where a casual worker has already agreed to an assignment. But if a casual worker declines a new assignment because of illness, you do not have to pay them.
If a part-time or hourly worker simply fails to turn up for work, you would not normally be required to pay them (unless they have an employment contract that says otherwise).