To start with, permanent and part time are not opposites. Permanent means your contract with an employer does not have an end point, you work until whenever, the job changes, you resign or you retire. The opposite is a fixed term contract, ie you start the job knowing you are working for 1 year only, or 3 months etc.

Part-time work is where you don't work a full working week, usually 35-37 hours per week. The opposite is full time work, where you do work something like Monday to Friday, 9-5.

So you can be permanent, full time, or permanent, part-time, or fixed-contract, full time, or fixed-contract part-time.

Whether you can change contract depends on the job. You need to speak to your HR staff and see if you can work on some agreed % of a full time equivalent. It might be possible to work at 80% fte, and that might mean 4 days per week, or 5 days per week, but leaving or starting at a different time.

There are also occasionally jobs that would work in a job share situation, where one person does the job Monday, Tuesday, Wednesday, and another person does the job Wednesday, Thursday and Friday (there is usually an overlap). However, most jobs don't really suit this.

To be able to ask your employer to consider a reduction in hours, I think you usually need to have worked there for a certain period of time, a year I think, and you usually have to be on a permanent, not a fixed contract. But HR would know the details and options.