Create a new site with a recent version or use the old one
This sounds like a great opportunity to kick-start a migration project to a supported and secure version of Django. Always use an LTS version: 2.2 is the most recent LTS version, 3.x isn't released yet.
You will need to copy the models from the old app and set managed = False, access the same db as the old project to modify the data with the new API.
Once your API can handle all the functionalities required by the website, it'll be quite easy to migrate the entire website.
Note the following:

Your models in the new project need to use the same database tables and columns, with the same field type. But nothing precludes you from renaming the actual models and fields (if you feel the old project used "bad" naming conventions), as long as you then specify db_column (on the field) and db_table (on the model) to let Django know the mapping to the existing database.
You can add new models and tables to your existing database (with managed = True) without affecting your old app. 

If at some stage you need to modify existing tables:

You might then consider switching to managed = True (and change your old app to managed = False), making sure that prior to that you'd have cleared the migrations table from the database and then run the migrations with fake set (so as to not actually migrate anything). I would do that if the old app is not maintained anymore.
If the old app is still maintained it's a bit more tricky. You would probably have to make the db changes directly in SQL.
Changes that won't affect the old app: Adding new columns.
Changes that will affect the old app: Anything that restricts constraints, or adds constraints, or changes the data type.
Changes that might affect the old app: Loosening constraints. This won't affect the existing data in your db, but it might trip validators of the old app that expect data in a certain format and get "invalid" data for the new data.
The project I'm working on is based on Django 1.4 and I can't change / update it actually.
I have to work on an API to modify data on that site.
I thought I would use Django Rest Framework to do it.
Would there be a problem if I start a new site from scratch using Django 3.X and the newest version of the DRF, using the same DataBase as the oldest site ?
I mean, if by using the API on the new site I change data on the DataBase, this is also going to be changed on the "old" ?
On the other side, I could keep working on the "old" site and use an old version of DRF but those version are less complete, less documented and less secure.
Thanks a lot