I agree. He's texting her in the middle of the night, he's spending all his time with her and he's spending nights at her house? Yes, he's cheating on you. It's the perfect storm - loss of a parent and reconnection with a "lost love".
And ask yourself - why are YOU doing all the care of his father? Why isn't HE spending all his time after work at his dad's?
Yes, he's depressed, and yes, people do crazy things in the fog of depression. But this is also a warning. You're not married to this man. Let's just say you get through this and he comes back to his senses and you marry him - what do you think happens ten or twenty years from now when the stresses of life hit again? He's showing you that he's not a good bet for a long term partner.
(And frankly, even if he's not sleeping with her - which I really think he is - he's not treating you with the respect and gratitude you deserve for everything you've done).
Let me tell you my story in a nutshell. ExH slept with old girlfriend the night before our wedding. 6 months later I find his journal where he's writing about her. I try to kick him out but he begs to stay. We repair our marriage (I think) and embark on life together. Three children and 14 years later he has an affair - children are devastated. I DB our marriage back together and have several more great years - then he approaches 50 and departs in a blaze of midlife crisis. We've now been divorced for 7 years and he's married to an Asian chick 19 years younger than him.
I blew past his red flags and lived to regret it. Don't make the mistake I did. My children suffered for the gamble I took on my ex.
(Your story also reminds me of my sister. She was married and had a two year old. Her mother-in-law was dying alone in another state. Even though this MIL was a terrible person who always treated my sister badly, my sister packed up her child and moved to take care of her in her final two months (my sister is a saint). Her husband stayed home to work and had an affair while she was gone! No, they're not married anymore - and he has shown himself to be a sociopath in the twenty some years since.)
Figure out where's best for you to be, start applying for jobs there (or see if you can get your old job back?) and start making plans to get your life back. Also start telling your BF "I have plans with friends from work tonight, youlll have to go to your dad's after work. " Then go out, even if it's just to a movie by yourself or to read a book in a coffeehouse.
He's showing you he can't be trusted in a time of stress. Life is full of stresses. If he can't make it through this without cheating, what do you think he'll do if you develop a chronic illness? Or have a child with a disability? You deserve a partner who adores you and is as good and reliable a person as you are. Don't settle.