There are several synonyms for the phrase "birthday card" that can be used depending on the occasion. One such synonym is "greeting card," which is a more general term that can be used for any type of occasion, such as weddings, baby showers, and anniversaries. Another synonym is "birthday greetings," which are often sent via email or text message. For a more formal or elegant occasion, "birthday invitation" is another synonym that can be used to indicate the party invitation aspect of the celebration. Ultimately, the synonym that is chosen will depend on the context and the style of the card or communication being used to convey the message.