Every bank will ask for a pps number - they are required to do so as this prevents money laundering and fraud.
I would be inclined to apply for the pps numbers, and I would do the following:
1. Apply before they have been in the country for three months (otherwise they may ask you for a payslip). IF you can apply before YOU have been in the country three months, this is even better.
2. Go with your parents to the pps office, take their passports, your passport (to show you are an EU citizen), your pps number and your birth cert (to show you are their child).
3. Take evidence that they have health insurance - the health insurance company should have sent them an insurance card and a letter outlining their policy details. The letter is proof of their address, the health insurance card shows they are covered by their health insurance for health care.
4. Take the letter from immigration which requested further details such as your mothers passport, as proof that immigration know your parents are in the country.
The Irish officials know that your parents are in Ireland, you have informed the immigration department, so there is nothing to lose by applying for a pps number, and they may succeed in getting one.
If the social welfare office query the need for a pps number, explain that you need it in order to get the dependent parent tax credit from revenue. Use that exact phrase 'dependent relative tax credit' and if the social welfare official has not heard of it suggest they look for it on the internet using google. I think that you have a right to get a pps number for them, in order to allow you to claim the tax advantages associated with supporting dependent parents/relatives.
You may also claim 'medical insurance relief' in respect of health insurance premiums paid in respect of a relative for whom you get the dependept relative tax credit.
To be honest, the tax stuff is not worth much, but it gives you a good reason to want a pps number for your parents.
You should apply for the dependent relative tax credit the moment you get a job, as revenue are likely to grant this quite easily, and if one government department grants a tax credit on the grounds that your parents are dependent, this supports your case that they are dependent.