First ask the broker if they have any information about the acquisition dates. Sometimes they have the information, even though they don't put it on the 1099-B. The fact that the broker classified some of the sales as short-term and some as long-term suggests that they know something about when the investments were acquired. Short-term means that the investment was acquired no more than a year before it was sold, so the broker should certainly know about those.
If you can't get any information from the broker, you can enter the word "Various" for the date acquired instead of an actual date. Or you can use your best guess as to the date. The actual date isn't critical, as long as it's consistent with whether the sale was short-term or long-term.
Maybe you can also dig around in your parents' papers. They might have kept some confirmations or other records of stocks that they bought.
