Error propagation methods for orbit parameters
ProfRob's comment is actually already the answer:

Unless you can linearise the model, there is no analytic error estimate.

I consider error estimation very important, so I would like to show a bit more details here: A way of estimating the error of a model is by Gaussian error propagation, see in particular the part of the Wikipedia entry on non-linear combinations. Let me briefly summarize the idea behind as I find the Wikipedia article not so easy to follow. I will also use different variable names so that they do not overlap with those in your question.
We start with a function $\varphi$ which depends on different variables $x_1, x_2, \ldots$, meaning $\varphi = \varphi(x_1, x_2, \ldots)$. We assume that we can linearize around a certain point $\tilde{\bf x} = (\tilde{x}_1, \tilde{x}_2, \ldots)$, which mainly means writing the function at that point as a Taylor series.
Then, we can determine the Gaussian (maximal) error in the vicinity of ${\bf \tilde{x}}$.
$$\left. \Delta \varphi \right|_{\bf \tilde{x}} = \left. \frac{\partial \varphi}{\partial x_1} \right|_{(\tilde{x}_1,\, \tilde{x}_2,\, \ldots )} \!\!\!\!\cdot \Delta x_1 + \left. \frac{\partial \varphi}{\partial x_2} \right|_{(\tilde{x}_1,\, \tilde{x}_2,\, \ldots )} \!\!\!\!\cdot \Delta x_2+\cdots$$
In this formula you calculate partial derivatives $\frac{\partial \varphi}{\partial x_i}$ with respect to each variable $x_i$ and plug in the values of the variables at the central point ${\bf \tilde{x}}$. The $\Delta x_i$ is the error estimate for each individual variable $x_i$.
How to we apply that to your problem where you have fitted an implicit equation? Since you excluded the Monte-Carlo approach (which is the usual method for this case), I suggest the following recipe:

Formulate explicit functions $a(x,y), b(x,y), \ldots f(x,y)$, e.g.
$$ a = - \frac{bxy + cy^2+ dx+ ey + f}{x^2}$$
Linearize the functions around $x_0,y_0$. The $x_0$ is called $x$ in your table, and respectively $y_0$ is in the $y$-column.
Determine $\Delta a, \ldots, \Delta f$ by partial differentiation.
Plugin in $\Delta x$ and $\Delta y$ as also given in your table.