  To achieve high-fidelity operations on a large-scale quantum computer, the
parameters of the physical system must be efficiently characterized with high
accuracy. For trapped ions, the entanglement between qubits are mediated by the
motional modes of the ion chain, and thus characterizing the motional-mode
parameters becomes essential. In this paper, we develop and explore physical
models that accurately predict both magnitude and sign of the Lamb-Dicke
parameters when the modes are probed {\it in parallel}. We further devise an
advanced characterization protocol that shortens the characterization time by
more than an order of magnitude, when compared to that of the conventional
method that only uses mode spectroscopy. We discuss potential ramifications of
our results to the development of a scalable trapped-ion quantum computer,
viewed through the lens of system-level resource trade offs.
