I am currently employed by my parents' company. As a tax saving option, they have decided to make me a partner. Does this mean I have to file a P45?
You can't fill in a P45 for yourself -- it has to be done by your previous employer. If you don't have a P45 from a previous employer, you have to fill in a P46, not a P45. But as a partner you may or may not count as a PAYE employee who needs a P45 or P46 -- I recommend you phone your tax office and ask.
A P45 is a form from a previous employer to say your employment has been terminated.
This is not required to start employment elsewhere.
However, it is typically required for your employer to send a P45 form to HMRC to indicate that you are no longer an employee (perhaps this is relevant to you, if you are effectively your own employer)
If you're moving on to work as an employee under different conditions/elsewhere, you certainly don't need to provide a P45 - one alternative is to fill out a P46 when you need to, but even that is excessive: your national insurance number (and perhaps tax code) will typically be enough for your new employer to set up your tax payments. P45 not required.
If you're no longer an employee, but self-employed/partner, it's likely you will need to send an annual tax return detailing your income. I am not an accountant so I can't advise you any further on this point.