While I can't comment on later periods, during the Roman period there were several areas that maintained quasi-independence, such as mountainous southern Turkey, interior Sardinia and northeast Spain. The Roman empire wasn't like a modern nation state, it didn't have polices of regional development or national integration, the "goals" of the state were essentially to extract taxes and prevent internal disturbance. It wasn't viewed as shameful or disturbing that parts of the empire did not live up to certain standards, nor were they perturbed by cultural differences (this is broadly true of most all pre-modern empires). Thus, if the rewards to be gained in taxation and stability by enforcing control were not worth the costs, the authorities tended not to bother.

A great trans-historical description of this from the perspective of these areas is James Scott's The Art of Not Being Governed. A case study of southern Turkey, which had similar dynamics as northeast Spain, is BD Shaw's "Bandit Highlands and Lowland Peace", which is a great read.

Adding on, were the Basque people ever persecuted or oppressed in ways similar to that of other European minorities such as the Jewish or Roma peoples?

The Romans never really occupied the Basque Country. They used it's roads more than anything else. As for the Basque language that's a little bit more difficult. The problem lies in the sheer number of dialects.The five main Basque dialects are Bizkaian, Gipuzkoan, and Upper Navarrese in Spain, and Navarrese–Lapurdian and Zuberoan(french basque country) it wasn't until ultra nationalist(the Basque kind) Sabino Arana decided to stanardise the language. It was made illegal under Franco and survived by people rarely speaking it in public. Now in the Basque country the language is having a huge revival through various educational and cultural means. He essentially created the basque alphabet such as replacing Spanish ch with an tx, c with a k and so on. A good example is the Basque wine Txakoli. In the rest of Spain it's spelt chacoli. This is seen throughout the Basque Country.

Basque History of the World - Mark Kurlansky

I would write more but am on my phone and have stubby thumbs

May I suggest taking this question to the nice people over at /r/linguistics as well? There are some very knowledgeable people over there who may not subscribe to /r/AskHistorians and who may well be able to cast light on the subject.

Amicable relations with the Romans followed by a staunch resistance to post-Roman invaders such as Visigoths and Franks. Geography aided the Basques here as they had intimate knowledge of the valleys which helped prepare ambushes (see the Battle of Roncesvalles).

The Basque people were a largely rural population and so the language was only superseded by Spanish in urban contexts (to this day, Basque fares well in areas like rural Gipuzkoa, whereas it is very rare in say, southern Araba around Vitoria-Gasteiz).

They were guaranteed certain rights by the Spanish crown via the fueros, or foruak which gave Basque people a certain degree of autonomy. It was not until the 19th century when the Basque language was oppressed by the Spanish state (see Moyano Law of 1857 for an early example).

Now, during the rule of Franco it was another story entirely. Basque language and culture was forcefully oppressed and measures were taken to ensure its extinction. In many ways such policies were successful, seeing as Basque today hovers somewhere around only half a million speakers and the majority of Basque Country is not fluent in the language. However with the introduction of the ikastolas (Basque schools) later in the 20th century children were still able to learn the language. Sweeping language reforms such as that led by Koldo Mitxelena in the 1960's standardized the Basque language (which was heavily fragmented by radically different dialects), ensuring a unified language which could be taught.