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Let's first denote the Euclidean transform by T_AB which transforms a point's coordinates in B frame p_B to the point's coordinates in A frame, p_A,

p_A = T_AB p_B.

Then let's clarify parent, child, target, and source frames.

Parent and child frames

If T_AB is phrased in terms of a ROS geometry_msgs/TransformStamped.msg, then the {A} frame is the parent frame and the {B} frame is the child frame. In ROS schematics, this means

$A \rightarrow B$

Target and source frames

To put T_AB in the context of a ROS Transformer, the {A} frame is the target frame and {B} the source frame in accordance with the lookupTransform API which transforms data from the source frame into the target frame.