Classical Bayesian persuasion assumes that an information designer can condition signals on the entire state. We propose persuasion by parts. In this model, the receiver's payoff from acting is $X = \sum_{i=1}^n X_i$ for independent coordinates $X_1,\ldots,X_n$ observed by separate sources. The designer commits in advance to a product experiment, in which each source $i$ can condition its signal only on $X_i$. Given the signals, the receiver computes her posterior mean $y$ of $X$ and acts with probability $g(y)$, where $g\colon \mathbb{R} \to [0,1]$ is a nondecreasing function. The designer aims to maximize the probability of action. Let $V_{\mathrm{prod}}(g)$ and $V_{\mathrm{joint}}(g)$ be the supremum probabilities of action under persuasion by parts and under joint persuasion, respectively. When the designer knows $g$, we show $$ V_{\mathrm{prod}}(g) \;\ge\; c\,V_{\mathrm{joint}}(g)^2 \qquad\text{for every nondecreasing } g\colon \mathbb{R} \to [0,1], $$ where $c > 0$ is an absolute constant independent of $n$ and the prior; the exponent $2$ is tight. When $g$ is unknown, we construct a product experiment $Π$ from the prior alone, whose probability of action $V_Π(g)$ satisfies $$ V_Π(g) \;\ge\; c_\varepsilon\,V_{\mathrm{joint}}(g)^{2 + \varepsilon} \qquad\text{for every } \varepsilon > 0 \text{ and every nondecreasing } g\colon \mathbb{R} \to [0,1], $$ where $c_\varepsilon > 0$ depends only on $\varepsilon$.

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