Can a parents allergies be passed on to their children?

Not exactly. The ability to react to certain proteins in an allergic way is passed on from parents to their children, but a specific allergy is not. So if a mom is allergic to pollen and the dad is allergic to fire ants, their child may develop allergies but it may be to a food instead. If 1 parent has allergies, the child is 50% likely to develop allergies, but it's a 75% chance if both parents are allergic.
Not Exactly. What is passed genetically from parent to child is the propensity to develop atopy (such as eczema, asthma, allergic rhinitis, food allergies) but usually not sensitization to specific allergens which the parent is allergic to. The exception, which is a controversial topic, is drug allergies, but this is usually dependent on metabolism of the drug which is inherited rather than the allergy.
Not really. Genetic predisposition allergies are transmitted however they don't always manifest.