They are two different things actually. Back4app is a managed service. They both manage database and parse server. If you ever decide to host your own server that means you have to manage your own server. Both database and parse server. You have to scale your app yourself. While there are guys in Back4app that get paid to do this for you.
One more tip: AWS is very complex environment. Be careful when using it. There are lots of pricing criteria. If you don’t have any experience with aws or you don’t know what you do, you may face some suprise bills.
Also Back4app owner is a member of this community. And he is very helpful. I believe he can give you the information you want. @davimacedo
I would tend to agree with @uzaysan and I would add that if you have little experience with AWS, managing backend infrastructure, scaling etc it may provide less headaches to go with a managed service. With both Back4app and Sashido you can spin up Parse Server and Parse Dashboard in minutes, they both provide easy configuration options, automatic scaling, backups and much more. Another advantage is that depending on your plan both providers offer support via email, chat etc.
However, I would also say that Parse Server is designed to remove complexity so if you have some AWS experience you should be able to get a basic Parse Server & Parse Dashboard setup fairly quickly. If you get stuck there are lots of existing resources in our docs, on this forum, on the parse-platform tag on Stack Overflow and in GitHub issues. Hosting yourself also allows you to make any customisations you wish for example you can maintain and deploy your own fork of Parse Server.
If I was you for your first project go with a managed solution and then as you get more comfortable with the Parse Platform you might want to try self-hosting.
@Tom mentioned some points. Also I forgot Sashido. They are also an option and their support team is very kind. They reply to your every question. Even some unrelated idiotic questions.(I tested that).
Anyway, If you wanna self host Parse server and doesn’t have any aws knowledge I recommend you to use some Vps server. I’m currently using a vps server and I installed parse server and run it in 3 minutes. They also comes with Storage bandwith etc. You dont have to worry about extra bills. Their price are fixed.