Touchscreen devices, ubiquitous in humans' day-to-day life, offer a promising avenue for animal enrichment. With advanced cognitive abilities, keen visual perception, and adeptness to engage with capacitive screens using dexterous tongues, parrots are uniquely positioned to benefit from this technology. Additionally, pet parrots often lack appropriate stimuli, supporting the need for inexpensive solutions using off-the-shelf devices. However, the current human-centric interaction design standards of tablet applications do not optimally cater to the tactile affordances and ergonomic needs of parrots. To address this, we conducted a study with 20 pet parrots, examining their tactile interactions with touchscreens and evaluating the applicability of existing HCI interaction models. Our research highlights key ergonomic characteristics unique to parrots, which include pronounced multi-tap behavior, a critical size threshold for touch targets, and greater effectiveness of larger targets over closer proximity. Based on these insights, we propose guidelines for tablet-based enrichment systems for companion parrots.