We are interested in parallelizing the Least Angle Regression (LARS) algorithm for fitting linear regression models to high-dimensional data. We consider two parallel and communication avoiding versions of the basic LARS algorithm. The two algorithms have different asymptotic costs and practical performance. One offers more speedup and the other produces more accurate output. The first is bLARS, a block version of LARS algorithm, where we update b columns at each iteration. Assuming that the data are row-partitioned, bLARS reduces the number of arithmetic operations, latency, and bandwidth by a factor of b. The second is Tournament-bLARS (T-bLARS), a tournament version of LARS where processors compete by running several LARS computations in parallel to choose b new columns to be added in the solution. Assuming that the data are column-partitioned, T-bLARS reduces latency by a factor of b. Similarly to LARS, our proposed methods generate a sequence of linear models. We present extensive numerical experiments that illustrate speedups up to 4x compared to LARS without any compromise in solution quality.