  We show that if F is a totally real field in which p splits completely and f
is a mod p Hilbert modular form with parallel weight 2<k<p, which is totally
ordinary at p and has tamely ramified Galois representation at all primes
dividing p, then there is a "companion form" of parallel weight k':=p+1-k. This
work generalises results of Gross and Coleman-Voloch for modular forms over Q.
