Referencing multiple tables from a single table and store additional data
I think what you're missing is a car_configuration table. This will have an FK to the cars table and have car_config_parts and car_config_tunings as children, like so:

create table car_configuration (
  config_id integer primary key,
  car_id integer references cars (car_id)
);

create table car_config_parts (
  config_id integer references car_configuration (config_id),
  part_id integer references parts (part_id),
  constraint car_config_part_pk primary key (config_id ,part_id)
);

create table car_config_tunings (
  config_id integer references car_configuration (config_id),
  tuning_id integer references tuning (tuning_id),
  constraint car_config_tuning_pk primary key (config_id, tuning_id)
);


You can then include the config_id as a reference in the timeset table (with no reference to cars). You'll need to ensure that you can't change the parts and tunings for a configuration or implement some form of versioning on this table to ensure you can reconstruct the exact details used to set a given time in the past.