Your question was discussed at Pseudorandom number generator · Issue #135 · fortran-lang/stdlib · GitHub , where Leon Foks mentioned his module Prng_Class
Class providing a pseudo-random number generator with an xorshift128+ or xorshift1024* generator that are both suitable for parallel applications. This is class is thread safe, and can be used in parallel without Prngs on different threads affecting each other.
The Scalable Parallel Random Number Generators Library (SPRNG) is in C and C++ but has a Fortran interface. A previous thread was RNG for MC simulations, where some vendor libraries were mentioned.