Conditions for refusal; appeal procedure. The Zoning Administrator or City Engineer shall not issue a land use permit for any property, the improvement of which might tend to interfere with the exterior lines of planned new streets, highways, parkways, parks or playgrounds, or the exterior lines of planned widening or extending of existing streets, highways, parkways, parks or playgrounds. Any person who feels aggrieved by the decision of the Zoning Administrator or City Engineer may appeal to the Zoning Board of Appeals, which has power in a specific case, by the vote of a majority of its members, to grant a permit for a building or such street, highway, parkway, park or playground, which will as little as practicable increase the cost of opening such street, highway, parkway, park or playground and such board may impose reasonable requirements as a condition of granting such permit, which requirements shall be designed to promote the health, convenience, safety or general welfare of the City. Such board shall refuse a permit where the applicant will not be substantially damaged by placing his building outside the planned street, highway, parkway, park or playground.