Liquidators on behalf of a now defunct bank have been paying my parents 6.5 per cent dividends but still owe over half their original amount.Can I as their sole heir now claim the remainder ?

Yes, but...

The executor(s) of the estate(s) does this, not you (unless you are the executor yourself). The process is that the executor(s) of your parents' estates must work with the bank to retitle the account in accordance with your parents' wills. This most likely will require that the executor provide death certificates and letters testimenatary to the bank to certify that the executor has authority to do this, then written instructions (probably notarized) to provide info about how to retitle. A call to the bank by the executor should help him get started - it's not dificult one the executor has proper authority documented.

It's a bit easier if your parents have establish payable-on-death instructions for the account - in this case all that's needed are that death certificates are provided to the bank, and then they will retitle the account per the instructions.