What resources are available for destitute parents?

Chambers, whatever you do do not start to support them. It will affect their ability to receive aid. There is no mention of their living arrangements, but I would get them on lists for senior housing - TODAY! It will provide housing at a percentage of their income and often some level of social services comes as part of the deal. Call the county office of aging and begin the process of applying for assistance. Food stamps (SNAP), medicaid, etc . What type of health care do they have?? She isn't eligible for Medicare until she is 65. But unless she is covered by his plan, she might quarlify for the affordable care package.
A bit of tough love here, unless there are extenuating circumstances, folks who are in their 60s with no savings and do not work don't bode well for a carefree retirement. It can be easy for those who are not fiscally responsible to drain others. Be cautious --- mother or not! You have 5 children to support and your own retirement to plan. If Mom and her husband run into food problems, get them alinged with a food pantry. Housing issues, get them into public/senior housing. Health coverage, ACA or medicaid. Heating /cooling sign them up for LIHEAP. If you have money to share, choose how you want to go about that. Bring a bag of groceries, for example. Good luck

Chambers I have found myself in similar circumstances with my Mum in recent years. To start you never mentioned why your Mum does not and has not worked much in her life. If she has an ongoing health issue that has left her with a chronic disability she may qualify for medicare before retirement age. If medicare deems her disabled she may also qualify for social security, either needs based or earned, depending on how little she has actually worked in her life.
The odd part about the system in the US is if you are penniless, as in the situation you believe your Mum is now in, or if you are Donald Trump the business man, you will be fine. The first, social services will take care of you and the second private healthcare companies will take care of you. It is the middle of the road, the regular people who struggle because it is far too expensive for an ordinary person to pay for long-term care.
Geewiz is absolutely correct in her post to you. Your priorities must be first and foremost to your husband and children. You can find post after post on this website alone detailing years of care provided for parents while marriages, families and bank accounts fall to the wayside. Guilt seems to be the driving force behind many caregivers decision making and planning. What many seem to forget is the part of the wedding ceremony when you "leave" your parents and "go" with your spouse to make a new life. Your spouse and children are your life and your children especially deserve a full and happy childhood. I am not dictating how and what that may look like for your family. All I am saying is that this is a huge decision that needs to be looked at and considered carefully, logically and sensibly rather than emotionally. You have a large family and finances are difficult to stretch even in a small one. That is why I advised that you try to keep emotions out of this discussion. What if you run through your finances in an effort to help them, leaving your family in need?
I know you said you feel obligated to help if you are able. I guess that is the real question that needs answering. Once you contact all of the agencies that have been suggested to you in the posts above and gathered all of the relevant information for your Mum and her partner, only then might you know what their future may look like. At that point you can sit down with your husband and discuss what, if anything, you can or want to do for them. After that discussion you can then sit down with your Mum and her partner and see where their thoughts have taken them and what they perceive their needs to be. From there you can explore any potential common ground or ideas to help them meet their own needs without you having to input any of your own finances.
Given the ages of your Mum and her partner no matter how your fact-finding mission turns out, in the worst case scenario you only have a handful of years until benefits would be available to help support and meet the needs of both of them. Hold onto that little piece of light if things get difficult in the meantime.
Remember through all of this that we teach others how to treat us. You are always in complete control of your own life and finances.
Never allow guilt to enter into any decision making process. I believe guilt is the most wasteful, energy-sucking emotion in the human experience. It serves no purpose but to hold a person stuck in one place in time. It does not change or heal the past situation from which it was created.
Wishing you peace as you traverse the decisions ahead.
With peace, gratitude and grace

They should apply for Medicaid. They will have to provide any and all bank statements going back several years, as well as taxes, etc. But your mom could end up with significant assistance at little or no cost to her.

Dear Chambers,

Good of you to look out for your mom and her husband. I have included the links to Medicaid and Medicare. I hope they help.

https://www.medicaid.gov/

https://www.medicare.gov/

They should apply for Medicaid. They will have to provide any and all bank statements going back several years, as well as taxes, etc. But your mom could end up with significant assistance at little or no cost to her.