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Choice B is correct. The equation of a line in the xy-plane can be written in slope-intercept form $y=mx+b$, where $m$ is the slope of the line and $\left(0,b\right)$ is its y-intercept. It’s given that the line passes through the point $\left(0,5\right)$. Therefore, $b=5$. It’s also given that the line is parallel to the graph of $y=7x+4$, which means the line has the same slope as the graph of $y=7x+4$. The slope of the graph of $y=7x+4$ is $7$. Therefore, $m=7$. Substituting $7$ for $m$ and $5$ for $b$ in the equation $y=mx+b$ yields $y=7x+5$.

Choice A is incorrect. The graph of this equation passes through the point $\left(0,0\right)$, not $\left(0,5\right)$, and has a slope of $5$, not $7$.

Choice C is incorrect. The graph of this equation passes through the point $\left(0,0\right)$, not $\left(0,5\right)$.

Choice D is incorrect. The graph of this equation passes through the point $\left(0,7\right)$, not $\left(0,5\right)$, and has a slope of $5$, not $7$.