Modern data sets, such as those in healthcare and e-commerce, are often derived from many individuals or systems but have insufficient data from each source alone to separately estimate individual, often high-dimensional, model parameters. If there is shared structure among systems however, it may be possible to leverage data from other systems to help estimate individual parameters, which could otherwise be non-identifiable. In this paper, we assume systems share a latent low-dimensional parameter space and propose a method for recovering $d$-dimensional parameters for $N$ different linear systems, even when there are only $T<d$ observations per system. To do so, we develop a three-step algorithm which estimates the low-dimensional subspace spanned by the systems' parameters and produces refined parameter estimates within the subspace. We provide finite sample subspace estimation error guarantees for our proposed method. Finally, we experimentally validate our method on simulations with i.i.d. regression data and as well as correlated time series data.