Let's rephrase the example sentence to the following:
Jane bought a radio for her parents to listen to the news.
The subject who will listen to the news is her parents. The subject of the action to buy is Jane. Only humans (or animals) can listen to something. Radio can't listen to anything. Without Jane's buying a radio, her parents won't be able to listen to the news. It clearly indicates a purpose of the action and the to-infinitive was used adverbially.
The best way to tell whether to-infinitive is used adjectivally or adverbially is to insert a subject of to-infinitive with a preposition for as shown above. The above sentence could be rephrased to:
Jane bought a radio so that her parents can listen to the news.
Let's take a look at the example in the comment:
I have bought a dog to frighten away burglars.
It could be rephrased to:
I have bought a dog who (the dog) can frighten away burglars.
The action of to frighten away will be done by the dog. Not by me. I would not have bought a dog if I had been able to frighten away burglars. This is closer to an adjectival usage. This is not to say it couldn't be interpreted as an adverbial usage.
Let's compare the following two sentences:
(1) He wants somebody to love him. vs (2) He wants somebody to love.
No. (1) clearly indicates an adjectival usage: He wants somebody who will (can) love him.
However, No. (2) is not clear. It could be either "He wants somebody (that) he will (can) love." or "He wants somebody so that he can love somebody or fall in love with somebody."
As you can see, it is ambiguous and it is meaningless to identify pedantically which usage is used. However, there is a clear case for the adverbial usage.
I went to the U.S. to study English.
Nobody will say this is an adjectival usage.