What happens if you minimize $D_{KL}(P_{parameters} || P_{data})$ under the Kullback-Leibler divergence?
(Azoury and Warmuth, 2001) have shown that minimizing the KL-divergence between two exponential family distributions is the same as minimizing the Bregman divergence that corresponds to their cumulant function, i.e.

$KL(P_{F,\theta_1} || P_{F,\theta_2}) = D_F(\theta_1 || \theta_2)$

For $f = \nabla F$, and $F^*$ the Fenchel conjugate of $F$, we get the following nice relationship for Bregman divergences:

$D_F(\theta_1 || \theta_2) = D_{F^*}(f(\theta_2) || f(\theta_1))$

See Lemma 4 in this paper, which also has references to other work that discusses this connection. The above can also be written without the $f$ transfers on $\theta$. So, there is a simple relationship between these two losses, i.e. between minimizing with $\theta_1$ as the first argument or with $\theta_2$ as the first argument.

Also, I feel I should mention that we can generally talk about maximizing likelihood in terms of Bregman divergences and P(data | params), rather than KL-divergences on densities. The above linked paper shows that

$\max_\theta log \ p(data | \theta) = \min_\theta D_F(\theta | data)$ 

The relationship above where we swapped the two variables of interest in the loss has a correspondence here:

$p(\theta | data) \ p(data) = p(data | \theta) \ p(\theta)$

giving

$log \ p(\theta | data) = log \ p(data | \theta) + log \ p(\theta) - log \ p(data)$

Since we typically do not put a prior on data, we find that maximizing the likelihood of $p(\theta | data)$ (typically called the MAP estimate) corresponds to maximizing likelihood of the data with a prior on the parameters. Correspondingly, maximum likelihood is like the MAP estimate, with a non-informative, uniform prior on $\theta$. 

K. S. Azoury and M. K. Warmuth. Relative loss bounds for on-line density estimation with the exponential family of distributions. 2001