What are the pros and cons of moving your parent/parents in with you in their elderly years to avoid them going to a care home?
If you look after them yourself you save the ruinous costs of care homes. You also have more control over their care regime. And it means you get to spend more time with your parents in their twilight years.
The downsides are that you have to provide the care or have someone else visiting your home to provide the care. If you can't handle it yourself and have home helpers visiting you're back to the chance that care will be scrimped and corners will be cut while also allowing strangers into your home with the associated risks of that. You also have to deal with your parents all the time, which may be very demanding depending on their needs and personality. A friend of mine has a grandmother in a nursing home - my memories of her was that she was a lot of fun to be around but now my friend's description is that she's just a mean old woman.
Depending on your family situation you may also have complications when parents finally pass. It's not unheard of for one child to provide extensive care for years only for their siblings to expect the estate to be divided absolutely equally with no consideration given to the sacrifices made by the one.
It can also be psychologically demanding if going out as a couple means hiring someone to sit in and watch the parents - even if it's saving the cost of a care home to some people it can feel like taking any time out gets ever-more expensive and they lose control of their own lives for an unknown time. If you can't even take a short break without figuring how to provide care for elderly relatives for that exact time that can become more draining than some people expect.
And of course moving them into your home requires that you live in a place big enough to house them and in a location they would be willing to live.