Active service during the First World War delayed his stage debut until 1922 at Devonshire Park Theatre, Eastbourne, with Charles Doran's Shakesperian company.
Educated at Xavier College and Bruges.
Following stints with the Huddersfield Repertory Company and the Abbey Theatre in Dublin, he made his London stage debut with the Liverpool Repertory Company at the Everyman in 1925.
His favourite role was often quoted as that of Charles Condamine in Noel Coward's farce "Blithe Spirit". It ran for 1,997 performances (form 1941), few of which Parker missed, establishing his reputation as one of Britain's foremost comedy actors. He himself contributed a sketch.
Made his first appearance on the New York stage in 1950 in Daphne Laurelo with Dame Edith Evans.