There are some options that appear as they might work, but they will not do quite the deal
Understanding languages will not work because it only affects the launcher (and does not translate to the launcher)
So that the wizard and the wizard can learn Understanding languages to at least understand what the dragon says, but they could not answer that way; they had to talk to the Paladin in common and ask the Paladin to answer the Dragon.
Hame of Understanding Languages has the same problem
the Hame of Understanding Languages is an unusual minor magic object. A permissive DM could allow you to get enough of these items (it's a minor magic item) for everyone to understand the Dragon; but you would always have the aforementioned problem that only the Paladin could answer the Dragon.
The spell Languages solves this problem much more effectively, even for a few hours a day:
This spell gives the creature you touch the ability to understand any spoken language it hears. In addition, when the target speaks, any creature that knows at least one language and can hear the target understands what it says.
–Languages, Player's Manual, 283
Once this spell is available, you would only need your cleric (or wizard or wizard if they want to choose the spell as a known spell) to cast it. Languages on the dragon, and for an hour at a time, the whole party would understand what the dragon says, and could also speak to the dragon and make it understand.
Faerie Dragons has an intelligence score of 14, which means that he should be relatively apt at learning new things. At the request of your SM, you can teach them the common language during the following 8 weeks of inactivity:
With enough free time and the services of an instructor, a character can learn a language or master a tool.
Resources. Receiving training in a language or tool usually takes at least ten weeks of work, but this time is reduced by a number of weeks of work equal to the character's intelligence modifier (a penalty of $ 10,000). Intelligence does not increase the time needed). The training costs 25 pounds a week.
–Revisited stop time, Xanathar's Guide to everything, pg. 134
Treat the fairy dragon as if it was a full-fledged character and treat the four members of the group as a single instructor, the fairy dragon should be able to learn to speak Common, if one has a enough time with the party.
Of course, the DM is not obliged to use the computer rules for NPCs, and this rule is considered an optional rule. The DM must therefore agree that it is an appropriate use of these rules. As a DM personally, I feel that it is a valid use of these rules.
You can also use the language training rules in the Player's Manual if your DM does not like the Xanathar review in Xanathar's Guide to Everything, although the DM must always agree that it is appropriate for an NPC uses these rules. If this is the case, the training takes about 35 weeks.
You can spend time between two adventures learning a new language or training with a set of tools. Your DM may allow additional training options.
First, you need to find an instructor willing to teach you. The DM determines how long it takes, and whether one or more capacity checks are required.
The training lasts 250 days and costs 1 gram per day. Once you have spent the necessary time and money, you are learning the new language or mastering the new tool.
–Training, Player's Manual, p. 187