Yes, you can, without adding a new column. You have to be careful to do it correctly in order to avoid corrupting the database, so you should completely back up your database before trying this.
for your specific example:
CREATE TABLE child(
id INTEGER PRIMARY KEY,
parent_id INTEGER,
description TEXT
);
--- create the table we want to reference
create table parent(id integer not null primary key);
--- now we add the foreign key
pragma writable_schema=1;
update SQLITE_MASTER set sql = replace(sql, 'description TEXT)',
'description TEXT, foreign key (parent_id) references parent(id))'
) where name = 'child' and type = 'table';
--- test the foreign key
pragma foreign_keys=on;
insert into parent values(1);
insert into child values(1, 1, 'hi'); --- works
insert into child values(2, 2, 'bye'); --- fails, foreign key violation
or more generally:
pragma writable_schema=1;
// replace the entire table's SQL definition, where new_sql_definition contains the foreign key clause you want to add
UPDATE SQLITE_MASTER SET SQL = new_sql_definition where name = 'child' and type = 'table';
// alternatively, you might find it easier to use replace, if you can match the exact end of the sql definition
// for example, if the last column was my_last_column integer not null:
UPDATE SQLITE_MASTER SET SQL = replace(sql, 'my_last_column integer not null', 'my_last_column integer not null, foreign key (col1, col2) references other_table(col1, col2)') where name = 'child' and type = 'table';
pragma writable_schema=0;
Either way, you'll probably want to first see what the SQL definition is before you make any changes:
select sql from SQLITE_MASTER where name = 'child' and type = 'table';
If you use the replace() approach, you may find it helpful, before executing, to first test your replace() command by running:
select replace(sql, ...) from SQLITE_MASTER where name = 'child' and type = 'table';