Yes, you can apply for Carer's Allowance even if you work part-time. Your earnings may affect the amount you can receive, so you might want to check your eligibility based on your specific situation.
If you're interested, it's a good idea to use online benefit calculators to see what you're entitled to regarding Carer's Allowance and other benefits. You can find them on various helpful sites linked in the published council information.
If you want to explore your options further or begin your application, find more information about Carer's Allowance here.