Sounds like you want
Tabulate Area (Spatial Analyst)—ArcGIS Pro | Documentation
But if you are using ArcGIS Pro and have write access to the property feature class, you can also do this with CalculateField.
Use Arcade as Expression Type and use this expression (change "TestPolygons" to the name of your forest feature class):
// load the forest fc
var forests = FeatureSetByName($datastore, "TestPolygons")
// get all forests intersecting the current feature
var intersecting_forests = Intersects($feature, forests)
// calculate the sum of all forest areas in the current feature
var forest_area = 0
for(var forest in intersecting_forests) {
forest_area += Area(Intersection($feature, forest))
}
// return that sum
return forest_area