Workforce Development Tax Credit Act of 2013 - Amends the Internal Revenue Code to allow a business-related tax credit for: (1) 50% of wages (up to $2,000) paid to an apprenticeship employee during an apprenticeship period, and (2) 40% of wages (up to $6,000) paid to such an employee during a post-apprenticeship period. Defines "apprenticeship employee" as any employee employed by an employer pursuant to an apprentice agreement registered with the Office of Apprenticeship of the Employment and Training Administration of the Department of Labor.