The board of directors of The Pittston Co. has elected a new president and vice chairman, the company said Friday. Joseph C. Farrell was elected president and chief operating officer, and David L. Marshall was elected vice chairman, the company said. Both had been executive vice presidents. Paul W. Douglas, chairman and chief executive officer, said the promotions were part of the transition plan for his scheduled retirement in September 1991. Pittston suffered a $10 million first quarter loss it blamed on a 10-month coal miners' strike and by poor performance in its air freight business. The Greenwich-based company is engaged in the mining and marketing of bituminous coal through the Pittston Coal Group, Inc. and Pyxis Resources Co.; in air freight services through Burlington Air Express, Inc.; in security transportation through Brink's Inc.; and in home security services through Brink's Home Security, Inc.