It is "periferia".
The link you provide to the "llevatilde" web page does not say that "perifería" must be accentuated according to any etymological reason, but just according to the Spanish ortography rules of accentuation. But the problem is that "perifería" just does not exist. In case of doubt, the RAE rules, so the link you provide to the DPD contains your answer.
Nonetheless, and as guifa pointed out in his comment, the fact that there IS an entry for periferia in the DPD makes us think that there must be somewhere where perifería is mainly used over periferia, or that it is a word that is spreading among some groups. Then we have two cases:
- You are a native Spanish speaker that has learnt to say perifería as it is the most used version in your place. Then you should stick to perifería as it reflects your way of speaking. It may be that in the future this version spreads and gets to be included in the DRAE.
- In any other case, you should stick to periferia as it is the accepted form by the RAE.
It seems that the form "perifería" has been adopted by some people as a false cultured word, as seen in the following text:
Pero el mal uso del lenguaje no es exclusivo de los pacientes, con el paso del tiempo, desde que iniciamos nuestra carrera los sanitarios tendemos a adoptar malas costumbres, modificando y en ocasiones dando nuevos sentidos a algunas palabras. Este hecho se afianza con el paso de los años, lo hacen nuestros maestros, los catedráticos, los adjuntos, nuestros tutores… y con la inercia terminamos haciéndolo nosotros hasta integrar las palabras en nuestro vocabulario habitual.
Tal es el caso de las siguientes “adopciones”:
– Estadío, donde debemos decir estadio. Es errónea la acentuación estadío, a pesar de ser frecuente en textos médicos.
– Perifería, donde debemos decir periferia. (periferia. ‘Contorno o alrededores’.) Es voz llana: [periféria]. Es incorrecta la forma con hiato perifería.
In fact, if you search "perifería" in Google (with quote marks to make the engine respect the tildes), the search engine gives you around 25900 results. But "periferia" gives you almost 20 million results.