Is it legal to provide a programming interface (API) to third party sites without their consent?

Question

API

You may have had to integrate third-party sites with your software product more than once. But often it turned out that these sites did not have open APIs. In such cases, I had to write a functional that would be able to log in and carry out some actions there (working with data / collecting information). And all this, of course, was carried out by parsing the dom-object. And here I had an idea, what if we make such a service that would take all this work on itself and provide users with the Restfull API (json / xml).
And the question itself is, how much will it be legal in a legal aspect?
Imagine that you are the owner of some promoted Internet resource and find out that some third company is distributing access to the resources of your site through a software interface. On the one hand, there is nothing illegal, data is accessed through the same channel and accounts (login & password), but not in the form (html) as originally intended by the owner of that very promoted resource. And this is not even a data collector (scrapping data), each user gets access to his data that is attached to his account. Although, after all, search engines (google / yandex) actually comb through the entire resource and provide links to them, cache them on their servers, and so on, and it seems that this does not contradict the laws.
I would like to clarify all these legal aspects, I would not want to spend months on development, and end up suing someone else.
I did not find such services, so I think since they are not there, maybe there are reasons:

• illegally - it is impossible so;
  
• crazy idea;
  

How do you think?

Answer 1

Depends on the owner of the site and the conditions under which he provides the service.
If it does, then it's illegal.

Answer 2

Imagine that you are the owner of a certain popular Internet resource and find out that a certain third company distributes access to the resources of your site through a software interface.

Just a ban in the so-called "rules" of the resource, non-persons. an agreement is not enough, it is still necessary to "bring it under the article", first of all, copyright infringement (copyrights, logos) suggests itself.
The idea is not crazy, but technically quite complex, requiring a number of personal characteristics and large investments, six-, seven-figure sums.