Measuring dark matter substructure within galaxy cluster halos is a fundamental probe of the {Lambda}CDM model of structure formation. Gravitational lensing is a technique for measuring the total mass distribution which is independent of the nature of the gravitating matter, making it a vital tool for studying these dark-matter-dominated objects. We present a new method for measuring weak gravitational lensing flexion fields, the gradients of the lensing shear field, to measure mass distributions on small angular scales. While previously published methods for measuring flexion focus on measuring derived properties of the lensed images, such as shapelet coefficients or surface brightness moments, our method instead fits a mass-sheet transformation invariant Analytic Image Model (AIM) to each galaxy image. This simple parametric model traces the distortion of lensed image isophotes and constrains the flexion fields. We test the AIM method using simulated data images with realistic noise and a variety of unlensed image properties, and show that it successfully reproduces the input flexion fields. We also apply the AIM method for flexion measurement to Hubble Space Telescope observations of A1689 and detect mass structure in the cluster using flexion measured with this method. We also estimate the scatter in the measured flexion fields due to the unlensed shape of the background galaxies and find values consistent with previous estimates.