sudare-me すだれ目

In manga and anime, sometimes a character's eyes are drawn simplified like horizontal parallel lines when they're relaxing, enjoying something or having fun. They're also drawn as parallel vertical lines sometimes.

Hirasawa Yui 平沢唯, Hirasawa Ui 平沢憂, example of closed eyes drawn using parallel horizontal lines.

Left: Hirasawa Yui 平沢唯
Right: Hirasawa Ui 平沢憂
Anime: K-On!!, Keion!! けいおん！！ (Season 2) (Episode 2)

Terminology

There doesn't seem to be a common term for this type of eye.

Exactly one page calls them sudare-me すだれ目, "window-blind eyes," because the parallel lines look like "window blinds made out of decorative wood," sudare 簾.

Everyone else seems to just call these by what they literally are:

yokosen-me
横線目
Horizontal-lines eyes. (also read ousen-me.)

However, due to how plurals work in Japanese, that can also mean the singular "horizontal LINE eyes," as in, just one single horizontal line, as opposed to multiple, and that's normally called ito-me 糸目 instead.

Furthermore, the same style is also used vertically.

tatesen-me
縦線目
Vertical-lines eyes. (also read juusen-me.)

And, likewise, that word can refer to eyes drawn with a single vertical line, which are known as konsento-me コンセント目 instead.

Examples

For reference, some examples:

Maki Oze 茉希尾瀬, example of sudare-me すだれ目.

Character: Maki Oze 茉希尾瀬
Anime: En'en no Shouboutai 炎炎ノ消防隊 (Episode 18)

The "red blood cell," sekkekkyuu 赤血球, AE3803, sighing.

Character: "red blood cell," sekkekkyuu 赤血球, AE3803
Anime: Hataraku Saibou はたらく細胞 (Episode 2)

tameiki
ため息
Sigh.

Yuno ゆの, example of eyes drawn with parallel vertical lines.

Character: Yuno ゆの
Anime: Hidamari Sketch, ひだまりスケッチ (Episode 1)

hatena maaku
はてなマーク
Question mark.

Assuming the above counts, we're talking about a texture—using multiple fine parallel lines instead of a single thick one to give the eye a soft feeling—and therefore, since the orientation of the lines shouldn't matter, even something like this should count too: