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Let $(x,y)$ be a point on the parabola. The distance from $(x,y)$ to the focus is $$\sqrt{(x-3{)}^2+(y-3{)}^2}.$$The distance from $(x,y)$ to the line $3x+7y-21=0$ is $$\frac{|3x+7y-21|}{\sqrt{3^2+7^2}}=\frac{|3x+7y-21|}{\sqrt{58}}.$$By definition of the parabola, these distances are equal. Hence,

$$\sqrt{(x-3{)}^2+(y-3{)}^2}=\frac{|3x+7y-21|}{\sqrt{58}}.$$Squaring both sides, we get $$(x-3{)}^2+(y-3{)}^2=\frac{(3x+7y-21{)}^2}{58}.$$This simplifies to $\boxed{49x^2-42xy+9y^2-222x-54y+603=0}.$