We establish $\ell^1$ error estimates for the infinite-dimensional multi-parameter Tikhonov regularization with $\ell^2$ and $\ell^0$ penalty terms.Lacking of interpolation inequalities within different sequences spaces(i.e.$\ell^2$,$\ell^0$ and $\ell^1$),one cannot directly obtain $\ell^1$ error estimates for the proposed multi-parameter Tikhonov functional where no $\ell^1$ information appears in the functional form.By building up the variational inequality in $\ell^1$ norm,we derive that both {\it a priori} and {\it a posteriori} parameter choice rules provide $\ell^1$ error estimates of optimal order under appropriate source conditions,and the corresponding constants can be verified in an explicit manner.In the latter case,we have implemented the classic and sequential discrepancy principles respectively.Finally an adopted prime-dual algorithm illustrates the sparsity promoting properties of the considered multi-parameter Tikhonov regularization.