High-dimensional sparse data emerge in many critical application domains such as healthcare and cybersecurity. To extract meaningful insights from massive volumes of these multi-dimensional data, scientists employ unsupervised analysis tools based on tensor decomposition (TD) methods. However, real-world sparse tensors exhibit highly irregular shapes and data distributions, which pose significant challenges for making efficient use of modern parallel processors. This study breaks the prevailing assumption that compressing sparse tensors into coarse-grained structures (i.e., tensor slices or blocks) or along a particular dimension/mode (i.e., mode-specific) is more efficient than keeping them in a fine-grained, mode-agnostic form. Our novel sparse tensor representation, Adaptive Linearized Tensor Order (${\sf ALTO}$), encodes tensors in a compact format that can be easily streamed from memory and is amenable to both caching and parallel execution. In contrast to existing compressed tensor formats,${\sf ALTO}$constructs one tensor copy that is agnostic to both the mode orientation and the irregular distribution of nonzero elements. To demonstrate the efficacy of${\sf ALTO}$, we accelerate popular TD methods that compute the Canonical Polyadic Decomposition (CPD) model across different types of sparse tensors. We propose a set of parallel TD algorithms that exploit the inherent data reuse of tensor computations to substantially reduce synchronization overhead, decrease memory footprint, and improve parallel performance. Additionally, we characterize the major execution bottlenecks of TD methods on multiple generations of the latest Intel Xeon Scalable processors, including Sapphire Rapids CPUs, and introduce dynamic adaptation heuristics to automatically select the best algorithm based on the sparse tensor characteristics. Across a diverse set of real-world data sets,${\sf ALTO}$outperforms the state-of-the-art approaches, achieving more than an order-of-magnitude speedup over the best mode-agnostic formats. Compared to the best mode-specific formats, which require multiple tensor copies,${\sf ALTO}$achieves$5.1\times$geometric mean speedup at a fraction (25% ) of their storage costs. Moreover,${\sf ALTO}$obtains$8.4\times$geometric mean speedup over the state-of-the-art memoization approach, which reduces computations by using extra memory, while requiring 14% of its memory consumption.