Forget your parents for the moment.
The person needing an understanding of what a PhD is (or should be, since so many "PhD" programmes today have corrupted the essence of it) is you yourself.
Below this there is the formal definition but let's just discuss what the essential human endeavour is first as that is what people reference their work to.
A PhD is research project on a topic you choose yourself and pursue essentially unassisted and "your own way" - i.e. using your own motivations, ideas, investigative approaches, evaluations and conclusions. While a supervisor is assigned to your programme, their contribution is more to provide a sounding board to you as you progress, discuss your findings and guard against work unlikely to be productive. While a good supervisor is often regarded as at least 60% of an adequate PhD submission, they will make a smaller contribution to a really good PhD - here the additional merit will come from your own hard work and your good fortune in topic choice and/or treatments applied by you.
The most important word in the above is the word motivations - the reasons you do the programme in the first place and hopefully the same reasons you stick with it as disappointments sap your morale.
You ask: Why not complete a PhD just for the sake of completing it ?.
If I understand this right, you see a PhD as nothing more than a mental test over 3 years with a qualification to teach or lead research at the end of it. While some PhD candidates do this and usually do a decent job of it, they seldom stay in research as it was never an end in itself. They often get well-enough paid jobs in the same sector or change to other sectors that recruit high academic achievers from all areas, e.g. managers in some industry, government administration, management consultants, MBA, top law schools, etc. If this is the high water mark of your motivation to do a PhD, then I agree with your parents that it is a poor use of time. But I don't think - and certainly hope - that this is why you are doing a PhD.
A better motivation would be to use the PhD as a starting point for a career in innovative industry, research and/or teaching in the same subject area. I think that all these types of careers - including research management at a later stage of your career - are all worthy justifications for entering a PhD programme. In fact, you would have little chance of getting a proper research job today without a PhD.
I can't say I can reference any articles that I find convincing. Of course, I know quite a few people whose career worked out as it should after their PhD in industry, government lab research and academia. But these are not people who want to be public poster examples for your parents ! Ask your professors if they know some past PhDs whose careers have gone as they should (or better) and refer to these talking to your folks.
MORE FORMAL DEFINITION
A PhD is a supervised research project done by an individual into an as-yet unanswered question in some subject area, viz. mathematics in your case. The research project must be:
Original and independent work - no "my supervisor suggested", etc although supervisor's constructive criticism and assistance with organization of work is allowed
Structured effectively - i.e. adequate work done investigating previous work on or around this topic, selection of potential approaches to the questions raised, evaluation of results obtained, conclusions drawn, etc
Adequately substantial to be regarded as a worthy addition to the field of knowledge on this topic
Defended in oral examination by an external academic well-versed in this area of research
Deemed worthy of publication in a peer-reviewed journal for that field of study, e.g. accepted for publication at some future point by he journal in question.