Vector of 2 Different Greek Symbols on Ggplot Grid Label

One option would be to first create separate columns for the phi and the theta values. Afterwards create the plotmath labels via e.g. paste0. To get right order of the panels convert the labels column to a factor:

library(ggplot2)
library(purrr)

ARMA11_MAE_NEWDAT$phi1 <- purrr::map_dbl(ARMA11_MAE_NEWDAT$psi, 1)
ARMA11_MAE_NEWDAT$theta1 <- purrr::map_dbl(ARMA11_MAE_NEWDAT$psi, 2)

ARMA11_MAE_NEWDAT$label <- paste0("varphi==", ARMA11_MAE_NEWDAT$phi1, "*textstyle(',')~", "theta==", ARMA11_MAE_NEWDAT$theta1)

ARMA11_MAE_NEWDAT$label <- factor(ARMA11_MAE_NEWDAT$label, levels = unique(ARMA11_MAE_NEWDAT$label))

ggplot(ARMA11_MAE_NEWDAT, aes(x = n, y = MAE, group = Methods)) +
  geom_point(aes(shape = Methods)) +
  geom_line(aes(linetype = Methods)) +
  xlab("Sample Size(n)") +
  ylab("RMSE") +
  ggplot2::facet_grid(sd ~ label, scales = "free_y", labeller = label_parsed) +
  scale_y_continuous(expand = c(0.0, 0.0)) +
  theme_bw(18) +
  theme(axis.text.x = element_text(angle = -90, vjust = 0.5)) +
  theme(axis.title.x = element_text(size = rel(1.00), angle = 0)) +
  theme(axis.title.y = element_text(size = rel(.55), angle = 90)) +
  theme(legend.title = element_text(size = 12), legend.text = element_text(size = 10))

DATA

library(reshape2)
library(tibble)

set.seed(199)
ARMA11_MAE_MBB_sd1_psi0.8 <-  runif(4, min = 0, max = 2)
ARMA11_MAE_MBBR_sd1_psi0.8 <- runif(4, min = 0, max = 2)
ARMA11_MAE_MBB_sd1_psi0.9 <-  runif(4, min = 0, max = 2)
ARMA11_MAE_MBBR_sd1_psi0.9 <- runif(4, min = 0, max = 2)
ARMA11_MAE_MBB_sd1_psi0.95 <-  runif(4, min = 0, max = 2)
ARMA11_MAE_MBBR_sd1_psi0.95 <- runif(4, min = 0, max = 2)

ARMA11_MAE_MBB_sd3_psi0.8 <-  runif(4, min = 2, max = 5)
ARMA11_MAE_MBBR_sd3_psi0.8 <- runif(4, min = 2, max = 5)
ARMA11_MAE_MBB_sd3_psi0.9 <-  runif(4, min = 2, max = 5)
ARMA11_MAE_MBBR_sd3_psi0.9 <- runif(4, min = 2, max = 5)
ARMA11_MAE_MBB_sd3_psi0.95 <-  runif(4, min = 2, max = 5)
ARMA11_MAE_MBBR_sd3_psi0.95 <- runif(4, min = 2, max = 5)

ARMA11_MAE_MBB_sd5_psi0.8 <-  runif(4, min = 5, max = 10)
ARMA11_MAE_MBBR_sd5_psi0.8 <- runif(4, min = 5, max = 10)
ARMA11_MAE_MBB_sd5_psi0.9 <-  runif(4, min = 5, max = 10)
ARMA11_MAE_MBBR_sd5_psi0.9 <- runif(4, min = 5, max = 10)
ARMA11_MAE_MBB_sd5_psi0.95 <-  runif(4, min = 5, max = 10)
ARMA11_MAE_MBBR_sd5_psi0.95 <- runif(4, min = 5, max = 10)

ARMA11_MAE_MBB_sd10_psi0.8 <-  runif(4, min = 10, max = 16)
ARMA11_MAE_MBBR_sd10_psi0.8 <- runif(4, min = 10, max = 16)
ARMA11_MAE_MBB_sd10_psi0.9 <-  runif(4, min = 10, max = 16)
ARMA11_MAE_MBBR_sd10_psi0.9 <- runif(4, min = 10, max = 16)
ARMA11_MAE_MBB_sd10_psi0.95 <-  runif(4, min = 10, max = 16)
ARMA11_MAE_MBBR_sd10_psi0.95 <- runif(4, min = 10, max = 16)

library(tibble)

ID <- rep(rep(c("10", "15", "20", "25"), 1), 1)
ARMA11_MAE_df1 <- data.frame(ID, ARMA11_MAE_MBB_sd1_psi0.8, ARMA11_MAE_MBBR_sd1_psi0.8, ARMA11_MAE_MBB_sd1_psi0.9, ARMA11_MAE_MBBR_sd1_psi0.9, ARMA11_MAE_MBB_sd1_psi0.95, ARMA11_MAE_MBBR_sd1_psi0.95, ARMA11_MAE_MBB_sd3_psi0.8, ARMA11_MAE_MBBR_sd3_psi0.8, ARMA11_MAE_MBB_sd3_psi0.9, ARMA11_MAE_MBBR_sd3_psi0.9, ARMA11_MAE_MBB_sd3_psi0.95, ARMA11_MAE_MBBR_sd3_psi0.95, ARMA11_MAE_MBB_sd5_psi0.8, ARMA11_MAE_MBBR_sd5_psi0.8, ARMA11_MAE_MBB_sd5_psi0.9, ARMA11_MAE_MBBR_sd5_psi0.9, ARMA11_MAE_MBB_sd5_psi0.95, ARMA11_MAE_MBBR_sd5_psi0.95, ARMA11_MAE_MBB_sd10_psi0.8, ARMA11_MAE_MBBR_sd10_psi0.8, ARMA11_MAE_MBB_sd10_psi0.9, ARMA11_MAE_MBBR_sd10_psi0.9, ARMA11_MAE_MBB_sd10_psi0.95, ARMA11_MAE_MBBR_sd10_psi0.95)

ARMA11_MAE_reshapp1 <- reshape2::melt(ARMA11_MAE_df1, id = "ID")
ARMA11_MAE_reshapp1 <- unique(ARMA11_MAE_reshapp1)

library(ggplot2)
library(tibble)

# Instead of using a character vector make your psi column a list column.

ARMA11_MAE_NEWDAT <- tibble::tibble(
  MAE = ARMA11_MAE_reshapp1$value,
  year = ARMA11_MAE_reshapp1$ID,
  n = rep(rep(c("10", "15", "20", "25"), each = 1), 24),
  Methods = rep(rep(c("MBB", "MBBR"), each = 4), 12),
  sd = rep(rep(c(1, 3, 5, 10), each = 24), 1),
  psi = rep(rep(list(c(0.5, 0.3), c(0.5, 0.4), c(0.35, 0.6)), each = 8), 4)
)

ARMA11_MAE_NEWDAT$sd <- factor(ARMA11_MAE_NEWDAT$sd, levels = ARMA11_MAE_NEWDAT$sd, labels = paste("sd ==", ARMA11_MAE_NEWDAT$sd))
ARMA11_MAE_NEWDAT$year <- factor(ARMA11_MAE_NEWDAT$year, levels = ARMA11_MAE_NEWDAT$year[1:4])
ARMA11_MAE_NEWDAT$n <- factor(ARMA11_MAE_NEWDAT$n, levels = ARMA11_MAE_NEWDAT$n[1:4])

Here is another solution by replacing the second part of the label after your function using the str_replace_all function from stringr package:

library(stringr)

# Create the labels
ARMA11_MAE_NEWDAT$psi <- purrr::map_chr(ARMA11_MAE_NEWDAT$psi, function(x) {
  paste(paste0("varphi", seq_along(x), "==", x), collapse = "*textstyle(',')~", sep = "")
})

ARMA11_MAE_NEWDAT$psi <- str_replace_all(ARMA11_MAE_NEWDAT$psi, c(varphi1="varphi ", varphi2 ="theta "))

Note that there is a typo in your labels function, the "varpsi" do not translate to a Greek letter, you can go with "varphi", "psi", "phi", ...