Detecting amyloid-β (Aβ) positivity is crucial for early diagnosis of Alzheimer's disease but typically requires PET imaging, which is costly, invasive, and not widely accessible, limiting its use for population-level screening. We address this gap by proposing a PET-guided knowledge distillation framework that enables Aβ prediction from MRI alone, without requiring non-imaging clinical covariates or PET at inference. Our approach employs a BiomedCLIP-based teacher model that learns PET-MRI alignment via cross-modal attention and triplet contrastive learning with PET-informed (Centiloid-aware) online negative sampling. An MRI-only student then mimics the teacher via feature-level and logit-level distillation. Evaluated across four MRI contrasts (T1w, T2w, FLAIR, T2*) and two independent datasets, our approach demonstrates effective knowledge transfer (best AUC: 0.74 on OASIS-3, 0.68 on ADNI) while maintaining interpretability and eliminating the need for clinical variables. Saliency analysis confirms that predictions focus on anatomically relevant cortical regions, supporting the clinical viability of PET-free Aβ screening. Code is available at github.com/FrancescoChiumento/pet-guided-mri-amyloid-detection.