The grounds for this refusal exceeds our normal canonical at UK visa refusal on V 4.2 a + c (and sometimes 'e')
That answer is relevant for what happened to your parents and the material there should not be duplicated here. So this answer deals with where they got YOU, the sponsor. They provide the rationale in the second dot point. In the larger view of things it is simply icing on the cake.
Had your statements been in perfect order and had they conceded that you have the capacity for sponsorship, the more serious grounds (specifically Paragraphs V 4.2 (a) + (c) + (e)) would have carried the refusal anyway.
V 4.3 (c)
This rule covers sponsorship for cases where applicants are in reduced circumstances or have some other economic disadvantage...
V 4.3 A visitor’s travel, maintenance and accommodation may be
provided by a third party where the decision maker is satisfied that
they: (c) can and will provide support to the visitor for the
intended duration of their stay.
The key words in this rule are 'can' and 'will'.
By 'can' they mean that the sponsor has to demonstrate their capacity for sponsorship by submitting all of the evidence that an applicant without sponsorship would submit. This includes bank statements, salary slips, employment contracts, and anything else related to financial readiness. PLUS proof that there is sufficient accommodation without violating the Housing Act PLUS if the housing is rented or provided by the council then evidence that the sponsor has permission to take in visitors. PLUS evidence of care if there are children or elderly involved. By 'will' they mean that you will collect them from the airport PLUS drop them off at the airport (and yes, we have refusals on those grounds in the archives here) PLUS provide all of the requisite maintenance and accommodation PLUS an attestation that you will be in the UK for the entire course of their visit.
Despite that most sponsors do not include all that stuff, the ECO has the option, but not the obligation to enforce any or all of those rules. That's what Paragraph V 4.3 (c) is telling you. And even if a sponsor is wealthy, a qualified solicitor will insist on covering all the bases.
It's worth remembering that they can absolutely nail a sponsor for funds parking just like they can for an applicant. And sadly, it's a credibility hit that's shared equally by both applicant and sponsor (even though it's not the applicant's fault).
Now for your specific questions...
and attach a statement from my sister saying she is happy for my mum
to use these funds for travelling to UK?
No. Bad idea. That means your sister becomes a co-sponsor and needs to establish her capacity for sponsorship just like you. See above for Paragraph V 4.3 (c).
Also, the other savings my parents have were not mentioned in previous
application as we didn't think it would be needed as my husband and I
are funding the visit - would this be a problem now?
Each time an application is made, it should start from square 1. Assumptions about what happened the last time or assumptions that the applicant has already established bona fides can lead to disaster. If there is evidence that is helpful and relevant, it should be included; whether or not it was included the last time doesn't matter.
My husband and I did attach our joint savings account statements which
showed a closing balance in excess of £7000 so they must have
It's known by everyone involved that VFS personnel in South Asia will remove evidence from applications some of the time. People are working on getting this fixed and have been doing that for quite some time. In the interim there's not much to be done unless you are represented. It's just part of WHAT IS for South Asian applicants.
Fortunately I don't think this was the case for you. I think they looked at it and determined that it is a savings account therefore not liquid therefore you do not intend to draw upon it therefore it doesn't qualify as sponsorship evidence therefore it is ignored. We don't know exactly what went through their minds but this is a conclusion they are entitled to make. So your claim that they overlooked it is probably specious.
Since there's a serious refusal on the books (V 4.2) and a complex sponsorship situation your next steps would be to get a so-called 'doc check'; that's where a member of the Law Society examines your evidence and points out gaps and how to fill them. It's not full on representation and the service level for a 'doc check' is substantially reduced, but your concerns are focused on what evidence to provide and how to best present it. Note that there is no legal aid for these things and so you'll need to pay.
The example I give for a 'doc check' is Colin Yeo. He charges somewhere in the neighbourhood of GBP 300+ for a doc check on a visit visa sponsor. Also note that there are qualified solicitors in Ilford (indeed, just walk down Green Lane!) who may charge less. You can also use the ILPA search engine to find a qualified solicitor with a practice area is South Asian visit applications.
You need to sort out your sponsorship evidence and the net is a horrible resource for this because sponsorship involves highly specific and personal stuff. Either get full representation for your parents or minimally a doc check. To apply again without those things is wildly adventurous.
Disclaimer: I have known Colin personally for about 12 years. It's only an example and there are hundreds out there. Note that receiving compensation of any sort for a referral is a violation of Sections 84 and 85 of the Immigration and Asylum Act 1999 (which provides the authority for the Commissioner's Code).