In ParseUser class there is method with name "logInInBackground" which provides a login API to ther user. This method contains three parameter "username", "password" and "LogInCallback" interface.

  ParseUser.logInInBackground("username", "password", new LogInCallback() {
      public void done(ParseUser user, ParseException e) {
        if (user != null) {
          // Hooray! The user is logged in.
          // User logged in successfully
         // call Home Activity
         Intent intent = new Intent(LoginActivity.this, HomeActivity.class);
         startActivity(intent);
        } else {
          // Signup failed. Look at the ParseException to see what happened.
          // Show Alert of exception
        }
      }
    });