1. I don’t recall about an obligation to reside in Lakewood. Sure, if we move to a community we have to conform to the accepted custom, but this does not mean we all must or should move to Lakewood.
2. You now got us very curious, and I imagine other readers too. What are those websites your family asks you to look up for them on your smartphone? are they simultaneously calling you on the cellphone? Why do you think they may be filtered out by a filter? (perhaps you can describe loosely if they are unsuitable to post.) Most of all why do they go through the hassle and expense, and waste your valuable time and their own valuable time, instead of looking up whatever they feel like? Also why should you lose respect for your parents? Ask a question along with all information, if it is permissible you will help, if not permissible you tell them “sorry but I can not help with this”, exactly like if they asked you to be mechalel shabbat, which is much worse than the internet.