Set expectations before the event
The single most useful thing a parent can do before a child's first tournament is separate the outcome from the effort. A child who understands that a loss doesn't erase a good tournament will play more freely — and usually perform better — than one who feels every game is a referendum on their ability.
Practical logistics
Make sure your child knows how a chess clock works before round one; fumbling with an unfamiliar clock under time pressure is a common and avoidable source of first-tournament stress. Pack snacks and water for between rounds, and plan for a full day — youth sections often run several rounds with only short breaks.
During the event
Parents generally aren't permitted in the playing hall once a round starts, which can be harder on the parent than the child. Use the break between rounds to keep the mood light rather than reviewing the last game move by move — that conversation is more useful with a coach later, away from the pressure of the next round.
After the tournament
A short debrief a day or two later, once the emotion has settled, tends to be far more useful than an immediate post-mortem. Ask what they noticed about their own play rather than leading with what went wrong — it builds the habit of self-analysis that will matter far more than any single result.