I use KeePass as well, but I don’t recommend it to non-technical folks 
. It’s not hard to use for people like us, perhaps, but it’s not simple enough for the average consumer. They need something that requires no extra effort or special knowledge. If even the slightest inconvenience arises, most people will simply go back to banging on the keyboard and then resetting their password every time they need to log in 
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As you mentioned, any file syncing system will do the job and this seems to be the most common approach today, but it typically must overwrite the “oldest” version of the database if both sides have changed. This could potentially cause data loss without careful usage (another reason I don’t recommend it to muggles). Though I rarely need the feature, I like to use the fine-grained synchronization built into the KDBX4 format.
To do this, I use the KeeAnywhere plugin and a Dropbox account that is used solely for storing the database; it has no connection with any of my other accounts and uses its own “master” password that I’ve practiced to the point that I should remember it until I forget my own name 
. I use KeePass2Android for mobile access as it natively supports Dropbox.
Everything I just said above is too difficult for your parents and probably your significant other as well. Thus, I recommend easier solutions. Most people already use Chrome, so that’s the easiest sell I’ve found. In my experience, people are simply not interested in using a password manager; it either sounds like too much hassle or too insecure (“putting your eggs all in one basket”). Rather than explain how all of their account “eggs” are already in one e-mail “basket”, I try to show them how Chrome can remember their passwords with no extra work. Even this rarely works 
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Though I use Firefox, I’m aware that Chrome is becoming bossy; Google sees itself as a sort of de facto “king of the Internet” and though many of its projects and decisions have been truly beneficial to all Internet users, I don’t want to live in a world where one company owns or controls my browser, my operating system (Android or ChromeOS), and much of the Internet itself. That’s why I’ve never switched to Chrome and never will; I’m stubborn like that .