Here is the source for saying Tehillim. I know it is difficult to accept, but it is unfortunately true. The Gemara and Shulchan Aruch say it is not allowed. Rabbi Abadi was very against it.
You can study Tehillim. That is different, but saying it to help sick people is prohibited.It is based on Christian Sources. It was not part of Judaism till the recent centuries. It is not halachicly permissible  

the more appropriate thing to do when someone you care about is sick, is to pray for them in the Amida/Shemoneh,esrey  or just say a short personal prayer in English between you and God. 

Our Lady’s Psalter