The parliament invested the new organisation ...... judicial authority.

The parliament invested the new organization with judicial authority.
In this context, "with" is the correct preposition to indicate the granting or conferring of authority or power upon something or someone. "With" is commonly used in this context to express the idea of providing or endowing something with a particular quality or ability. Therefore, "with" is the correct choice to complete the sentence, accurately conveying the action of investing the new organization with judicial authority.