Quantum signal processing (QSP) provides a methodology for constructing a polynomial transformation of a linear operator encoded in a unitary. Applied to an encoding of a density matrix $\rho$, QSP enables the evaluation of nonlinear functions of the form $\tr(P(\rho))$ for a polynomial $P(x)$, which encompasses relevant properties like entropies and fidelity. However, QSP is a sequential algorithm: implementing a polynomial of degree $d$ necessitates a circuit depth $O(d)$, which can be prohibitive on near-term quantum computers whose depths are limited by noise. Here, we show how to reduce the depth of these property estimation algorithms by developing Parallel QSP. Our algorithm parallelizes the computation of $\tr (P(\rho))$ over $O(k)$ systems and reduces the circuit depth to $O(d/k)$, thus illustrating a depth-width trade-off for QSP. This is achieved by factorizing $P(x)$ into a product of $O(k)$ smaller polynomials of degree $O(d/k)$, which are each implemented in parallel with QSP, and subsequently multiplied together with a generalized SWAP test to reconstruct $P(x)$. This furnishes a property estimation algorithm with a markedly reduced circuit depth $O(d/k)$, at the expense of increasing the number of measurements by a factor $2^{O(k)}$. We characterize the polynomials that can be implemented by parallel QSP by appealing to the fundamental theorem of algebra, and demonstrate application of our algorithm to entropy estimation and entanglement spectroscopy.