§ IDescription
Parakolouthein literally means "to follow alongside," from παρά ("alongside, at") + ἀκολουθέω ("to follow"). In Stoic technical use, especially in Epictetus, it designates a specific faculty (παρακολουθητικὴ ἰδιότης) of the rational part of the soul: this faculty alone, among all the ψυχικαὶ δυνάμεις (psychic capacities), is able to track itself — to notice its own acts, evaluate them, reject some and confirm others. Sight sees colours, hearing hears sounds, but not the reverse; sight does not see sight. Yet reason sees both sight and hearing, and its own activity besides. This reflexive capacity is the structural foundation of Stoic ethics: responsibility for one's assents (συγκατάθεσις) would be impossible without the capacity to track them.
As a practice, this is neither auto-psychoanalysis nor "immersion in the self" in the Romantic sense, but a sober, attentive accompaniment of one's own stream: which impressions are coming in, which assents I am granting, which impulses arise from these. The aim is not to describe oneself but to intervene early — to notice a false assent at the moment it occurs, rather than after the fact, by its consequences. It is the inner version of attention: if προσοχή is turned toward the action at hand, parakolouthēsis is turned toward the present movement of one's own soul.
§ IITechnique
(1) Hold a background "second gaze" on your own state in everything you do: not just what I am thinking / doing, but what I am thinking in the moment, what I like about this thought, which passion is being mixed in. (2) When a strong impression appears, pause: what is this, do I assent to it, what will follow. (3) Regular "snapshots": in the morning, what I am bringing into the day; in the evening, what has happened in my soul today (Seneca's form, De ira III 36: "cotidie apud me causam dico" — "every day I plead my case before myself"). (4) Do not confuse with anxious self-scrutiny: parakolouthēsis is cold, not emotional. It is a diagnostic instrument, not an occasion for feelings about oneself.