$\newcommand{\parcir}[2]{\frac{\partial^R #1}{\partial #2}}$ $\newcommand{\parcil}[2]{\frac{\partial^L #1}{\partial #2}}$ $\newcommand{\vprcir}[2]{\frac{\overleftarrow{\partial} #1}{\partial #2}}$ $\newcommand{\vprcil}[2]{\frac{\overrightarrow{\partial} #1}{\partial #2}}$ $\newcommand{\corcc}[1]{\left[#1\right]}$ $\newcommand{\gcorch}[1]{\left\lbrace#1\right\rbrace}$ $\newcommand{\parn}[1]{\left(#1\right)}$ $\newcommand{\parci}[2]{\frac{\partial #1}{\partial #2}}$
Working with Grassmann parity, the graded Jacobi identity is
\begin{equation} \gcorch{\gcorch{f_1,f_2}, f_3} + (-1)^{\epsilon_{f_1}\parn{\epsilon_{f_2}+\epsilon_{f_3}}}\gcorch{\gcorch{f_2, f_3}, f_1} + (-1)^{\epsilon_{f_3}\parn{\epsilon_{f_1}+\epsilon_{f_2}}}\gcorch{\gcorch{f_3,f_1}, f_2} = 0. \end{equation} Trying to do a proof, I have considered that: The generalized Poisson bracket is $\gcorch{f, g} = \parcir{f}{z^A}\Gamma^{AB}\parcil{f}{z^B}$, the left and right derivative are related by
\begin{equation} \parcir{f}{z^A} = (-1)^{\epsilon_{A}(\epsilon_{f} + 1)}\parcil{f}{z^A}, \hspace{0.2cm} \parcil{f}{z^A} = (-1)^{\epsilon_{A}(\epsilon_{f} + 1)}\parcir{f}{z^A}, \end{equation}
and the parity of the derivative (in general) is $\epsilon\parn{\parci{f}{z^A}}=\epsilon_f + \epsilon_A$. So
\begin{gather} \gcorch{\gcorch{f_1,f_2}, f_3} = \parcir{\gcorch{f_1, f_2}}{z^A} \Gamma^{AB} \parcil{f_3}{z^B} \\ = \parcir{}{z^A}\parn{\parcir{f_1}{z^C} \Gamma^{CD} \parcil{f_2}{z^D}} \Gamma^{AB} \parcil{f_3}{z^B} \\ = \gcorch{\parcir{f_1}{z^C} \Gamma^{CD} \corcc{\parcir{}{z^A}\parn{\parcil{f_2}{z^D}}} + (-1)^{\epsilon_A(\epsilon_{f_2}+\epsilon_D)} \corcc{\parcir{}{z^A}\parn{\parcir{f_1}{z^C}}}\Gamma^{CD} \parcil{f_2}{z^D}} \Gamma^{AB} \parcil{f_3}{z^B} \\ =\gcorch{\parcir{f_1}{z^C} \Gamma^{CD} \corcc{\parcir{}{z^A}\parn{\parcil{f_2}{z^D}}}}\Gamma^{AB} \parcil{f_3}{z^B} + (-1)^{\epsilon_A(\epsilon_{f_2}+\epsilon_D)} \gcorch{\corcc{\parcir{}{z^A}\parn{\parcir{f_1}{z^C}}}\Gamma^{CD} \parcil{f_2}{z^D}} \Gamma^{AB} \parcil{f_3}{z^B} \\ =\gcorch{\parcir{f_1}{z^C} \Gamma^{AB} \corcc{\parcir{}{z^A}\parn{\parcil{f_2}{z^D}}}}\Gamma^{CD} \parcil{f_3}{z^B} + (-1)^{\epsilon_A(\epsilon_{f_2}+\epsilon_D)} \gcorch{\corcc{\parcir{}{z^A}\parn{\parcir{f_1}{z^C}}}\Gamma^{AB} \parcil{f_2}{z^D}} \Gamma^{CD} \parcil{f_3}{z^B} \\ =\gcorch{\parcir{f_1}{z^C} \Gamma^{AB} \corcc{\parcir{}{z^A}\parn{\parcil{f_2}{z^D}}}}\Gamma^{CD} \parcil{f_3}{z^B} \\ + (-1)^{\epsilon_A(\epsilon_{f_2}+\epsilon_D)+(\epsilon_{f_3}+\epsilon_B)(\epsilon_{f_2}+\epsilon_D)} \gcorch{\corcc{\parcir{}{z^A}\parn{\parcir{f_1}{z^C}}} \Gamma^{AB} \parcil{f_3}{z^B} }\Gamma^{CD} \parcil{f_2}{z^D} \\ = \gcorch{\parcir{f_1}{z^C} \Gamma^{AB} \corcc{\parcir{}{z^A}\parn{\parcil{f_2}{z^D}}}}\Gamma^{CD} \parcil{f_3}{z^B} \\ + (-1)^{\epsilon_A(\epsilon_{f_2}+\epsilon_D)+(\epsilon_{f_3}+\epsilon_B)(\epsilon_{f_2}+\epsilon_D)+\epsilon_C \epsilon_A} \gcorch{\corcc{\parcir{}{z^C}\parn{\parcir{f_1}{z^A}}} \Gamma^{AB} \parcil{f_3}{z^B} }\Gamma^{CD} \parcil{f_2}{z^D} \\ = (-1)^{\epsilon_A(\epsilon_{f_2}+\epsilon_D + 1)} \gcorch{\parcir{f_1}{z^C} \Gamma^{AB} \corcc{\parcil{}{z^A}\parn{\parcil{f_2}{z^D}}}}\Gamma^{CD} \parcil{f_3}{z^B} \\ + (-1)^{\epsilon_A(\epsilon_{f_2}+\epsilon_D)+(\epsilon_{f_3}+\epsilon_B)(\epsilon_{f_2}+\epsilon_D)+\epsilon_C \epsilon_A} \gcorch{\corcc{\parcir{}{z^C}\parn{\parcir{f_1}{z^A}}} \Gamma^{AB} \parcil{f_3}{z^B} }\Gamma^{CD} \parcil{f_2}{z^D} \\ = (-1)^{\epsilon_A(\epsilon_{f_2} + 1)} \gcorch{\parcir{f_1}{z^C} \Gamma^{AB} \corcc{\parcil{}{z^D}\parn{\parcil{f_2}{z^A}}}}\Gamma^{CD} \parcil{f_3}{z^B} \\ + (-1)^{\epsilon_A(\epsilon_{f_2}+\epsilon_D)+(\epsilon_{f_3}+\epsilon_B)(\epsilon_{f_2}+\epsilon_D)+\epsilon_C \epsilon_A} \gcorch{\corcc{\parcir{}{z^C}\parn{\parcir{f_1}{z^A}}} \Gamma^{AB} \parcil{f_3}{z^B} }\Gamma^{CD} \parcil{f_2}{z^D} \\ = (-1)^{\epsilon_A(\epsilon_{f_2} + 1)+(\epsilon_{f_1}+\epsilon_C)(\epsilon_D + \epsilon_{f_2} + \epsilon_A)} \gcorch{\corcc{\parcil{}{z^D}\parn{\parcil{f_2}{z^A}}} \Gamma^{AB} \parcir{f_1}{z^C}}\Gamma^{CD} \parcil{f_3}{z^B} \\ + (-1)^{\epsilon_A(\epsilon_{f_2}+\epsilon_D)+(\epsilon_{f_3}+\epsilon_B)(\epsilon_{f_2}+\epsilon_D)+\epsilon_C \epsilon_A} \gcorch{\corcc{\parcir{}{z^C}\parn{\parcir{f_1}{z^A}}} \Gamma^{AB} \parcil{f_3}{z^B} }\Gamma^{CD} \parcil{f_2}{z^D} \\ = (-1)^{(\epsilon_{f_1}+\epsilon_C)(\epsilon_D + \epsilon_{f_2} + \epsilon_A) + (\epsilon_{f_1}+\epsilon_C)(\epsilon_{f_3}+\epsilon_B)} \gcorch{\corcc{\parcil{}{z^D}\parn{\parcir{f_2}{z^A}}} \Gamma^{AB} \parcil{f_3}{z^B}} \Gamma^{CD} \parcir{f_1}{z^C} \\ + (-1)^{\epsilon_A(\epsilon_{f_2}+\epsilon_D)+(\epsilon_{f_3}+\epsilon_B)(\epsilon_{f_2}+\epsilon_D)+\epsilon_C \epsilon_A} \gcorch{\corcc{\parcir{}{z^C}\parn{\parcir{f_1}{z^A}}} \Gamma^{AB} \parcil{f_3}{z^B} }\Gamma^{CD} \parcil{f_2}{z^D} \\ = (-1)^{(\epsilon_{f_1}+\epsilon_C)\corcc{(\epsilon_D + \epsilon_{f_2} + \epsilon_A) (\epsilon_{f_3}+\epsilon_B + 1) +(\epsilon_{f_3} + \epsilon_B)}} \parcir{f_1}{z^C} \Gamma^{CD} \gcorch{\corcc{\parcil{}{z^D}\parn{\parcir{f_2}{z^A}}} \Gamma^{AB} \parcil{f_3}{z^B}} \\ + (-1)^{\epsilon_A(\epsilon_{f_2}+\epsilon_D)+(\epsilon_{f_3}+\epsilon_B)(\epsilon_{f_2}+\epsilon_D)+\epsilon_C \epsilon_A} \gcorch{\corcc{\parcir{}{z^C}\parn{\parcir{f_1}{z^A}}} \Gamma^{AB} \parcil{f_3}{z^B} }\Gamma^{CD} \parcil{f_2}{z^D}. \end{gather} From here I suppose that zeros can be added to be able to group terms, however, it is not clear to me what it is convenient to add.