Should I open a new checking account at another bank?

There's no problem in having multiple checking accounts, and separating out the activities of building the house for your parents is probably a good idea. (I've got three checking accounts.)

Presumably you'll be depositing Large checks, so an online bank is out of the question.

Next is monthly fees. Unless you want to park $1,500 in the account and have it sit there doing nothing, Chase will charge you $12/month for the pleasure of having an account with them. BoA, WF and Citibank probably have similar policies.

Thus, I'd look for a local bank or credit union that doesn't have monthly fees.

(Changing subjects: does Chase charge you $12 every month?)

Contact your bank or credit union to see if you can have two checking accounts.

The credit union I use has allowed a "secondary checking account". That allowed me to run all my income and expenses though the 2nd checking account for my rental property. There was zero cost, and the 2nd account made tracking everything easier. If there was a reason to move money from the personal account to the 2nd checking account, it was very easy to do.

If the existing Chase account can't be accessed from a local branch, then there is a benefit to having the mortgage money in an account that is easy access from a local branch to meet your requirements for cash deposits. If you go this route it is also possible to move money between different financial institutions because most of them offer the ability to link accounts.

I've heard that it's generally better to have separate checking accounts at different banks. But I don't know what reason behind it is, and what I've heard might be wrong.

In the United States the coverage through FDIC or NCUA is for $250K, and it can be higher based on ownership of accounts. As long as the account is FDIC or NCUA insured and the sum of your Chase accounts is under that limit you shouldn't be worried about reputation. Some people spread their money across banking companies if they are concerned about the 250K limit, or they are concerned about tying up all their money in one bank, and having no access if that bank has an issue.

Of course customer service ratings are a different matter and you have to do your research.

You can have as many checking accounts as you like but it makes no sense to have multiple accounts unless you have a specific need for segregating assets such as business and pleasure or as you mentioned, mortgage and personal.

If possible, you want to avoid monthly fees. They can really add up. I have two free checking accounts at different banks with no minimum (15+ and 25+ years) but I don't know if that is still available for new accounts. If you can find a bank that offers that, for convenience, it makes sense to have both checking accounts at one bank.

Convenience comes in all sizes. When my father got into his mid 70's, while he was capable of balancing a checkbook, he liked simplification so he used one bank's checking account for one month and then used another for the next month. :->)

Another possibility might be to utilize a high yield money market account for the mortgage and associated deposits, and if so inclined, link it to your current Chase account. Since mortgage payments are monthly payments, the 6 transaction per month limit might not be a problem, plus you'll earn a decent interest rate (2+ pct).